2 derivative of a complex function. Complex derivatives

The operation of finding a derivative is called differentiation.

As a result of solving the problems of finding derivatives for the simplest (and not very simple) functions by defining the derivative as the limit of the ratio of the increment to the increment of the argument, a table of derivatives and precisely defined rules of differentiation appeared. The first in the field of finding derivatives were Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716).

Therefore, in our time, in order to find the derivative of any function, it is not necessary to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but you just need to use the table of derivatives and the rules of differentiation. The following algorithm is suitable for finding the derivative.

To find the derivative, you need an expression under the stroke sign disassemble simple functions and determine what actions (product, sum, quotient) these functions are linked. Further, the derivatives of elementary functions are found in the table of derivatives, and the formulas for derivatives of the product, sum and quotient are found in the rules of differentiation. Derivative table and rules of differentiation are given after the first two examples.

Example 1. Find the derivative of a function

Solution. From the rules of differentiation, we find out that the derivative of the sum of functions is the sum of the derivatives of functions, i.e.

From the table of derivatives we find out that the derivative of the "x" is equal to one, and the derivative of the sine is equal to the cosine. We substitute these values ​​into the sum of derivatives and find the derivative required by the condition of the problem:

Example 2. Find the derivative of a function

Solution. We differentiate as the derivative of the sum, in which the second term with a constant factor, it can be taken outside the sign of the derivative:

If there are still questions about where what comes from, they, as a rule, become clearer after familiarization with the table of derivatives and the simplest rules of differentiation. We are going to them right now.

Derivative table of simple functions

1. Derivative of a constant (number). Any number (1, 2, 5, 200 ...) that is in the function expression. Always zero. This is very important to remember, as it is required very often.
2. Derivative of the independent variable. Most often "x". Always equal to one. This is also important to remember for a long time.
3. Derivative degree. When solving problems, you need to transform non-square roots into a degree.
4. Derivative of a variable to the power of -1
5. Derivative square root
6. Derivative of sine
7. Derivative of the cosine
8. Derivative of the tangent
9. Derivative of the cotangent
10. Derivative of the arcsine
11. Derivative of the arccosine
12. Derivative of the arctangent
13. Derivative of the arc cotangent
14. Derivative of the natural logarithm
15. Derivative of the logarithmic function
16. Derivative of the exponent
17. Derivative of the exponential function

Differentiation rules

1. Derivative of the sum or difference
2. Derivative of the work
2a. Derivative of an expression multiplied by a constant factor
3. Derivative of the quotient
4. Derivative of a complex function

Rule 1.If functions

differentiable at some point, then at the same point the functions

moreover

those. the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions.

Consequence. If two differentiable functions differ by a constant term, then their derivatives are equal, i.e.

Rule 2.If functions

differentiable at some point, then at the same point their product is also differentiable

moreover

those. the derivative of the product of two functions is equal to the sum of the products of each of these functions by the derivative of the other.

Corollary 1. The constant factor can be moved outside the sign of the derivative:

Corollary 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of the factors by all the others.

For example, for three factors:

Rule 3.If functions

differentiable at some point and , then at this point it is differentiable and their quotientu / v, and

those. the derivative of the quotient of two functions is equal to the fraction, the numerator of which is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the previous numerator.

Where what to look for on other pages

When finding the derivative of the product and the quotient in real problems, it is always necessary to apply several differentiation rules at once, so there are more examples of these derivatives in the article"Derivative of a work and a particular function".

Comment. Do not confuse a constant (that is, a number) as a summand and as a constant factor! In the case of a term, its derivative is equal to zero, and in the case of a constant factor, it is taken out of the sign of the derivatives. it typical mistake, which occurs at the initial stage of studying derivatives, but as several one- or two-component examples are already solved, the average student no longer makes this mistake.

And if, when differentiating a work or a particular, you have a term u"v, in which u- a number, for example, 2 or 5, that is, a constant, then the derivative of this number will be equal to zero and, therefore, the entire term will be equal to zero (this case is analyzed in Example 10).

Another common mistake is the mechanical solution of a derivative of a complex function as a derivative of a simple function. That's why derivative of a complex function a separate article is devoted. But first, we will learn to find the derivatives of simple functions.

Along the way, you can not do without expression transformations. To do this, you may need to open the tutorials in new windows Actions with powers and roots and Actions with fractions .

If you are looking for solutions to derivatives of fractions with powers and roots, that is, when a function looks like , then follow the lesson Derivative of the Sum of Fractions with Powers and Roots.

If you have a task like , then your lesson "Derivatives of simple trigonometric functions".

Step by step examples - how to find the derivative

Example 3. Find the derivative of a function

Solution. We determine the parts of the function expression: the whole expression represents the product, and its factors are sums, in the second of which one of the terms contains a constant factor. We apply the rule of product differentiation: the derivative of the product of two functions is equal to the sum of the products of each of these functions by the derivative of the other:

Next, we apply the rule for differentiating the sum: the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions. In our case, in each sum, the second term with a minus sign. In each sum we see both an independent variable, the derivative of which is equal to one, and a constant (number), the derivative of which is equal to zero. So, "x" for us turns into one, and minus 5 - into zero. In the second expression, "x" is multiplied by 2, so we multiply two by the same unit as the derivative of "x". We get the following values ​​of the derivatives:

We substitute the found derivatives into the sum of the products and obtain the derivative of the entire function required by the condition of the problem:

And you can check the solution of the problem for the derivative on.

Example 4. Find the derivative of a function

Solution. We are required to find the derivative of the quotient. We apply the formula for differentiating the quotient: the derivative of the quotient of two functions is equal to a fraction, the numerator of which is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the previous numerator. We get:

We have already found the derivative of the factors in the numerator in Example 2. Let's not forget that the product that is the second factor in the numerator in the current example is taken with a minus sign:

If you are looking for solutions to problems in which you need to find the derivative of a function, where there is a continuous heap of roots and powers, such as, for example, then welcome to class "Derivative of the sum of fractions with powers and roots" .

If you need to learn more about the derivatives of sines, cosines, tangents and others trigonometric functions, that is, when the function looks like , then your lesson "Derivatives of simple trigonometric functions" .

Example 5. Find the derivative of a function

Solution. In this function, we see a product, one of the factors of which is the square root of the independent variable, the derivative of which we familiarized ourselves with in the table of derivatives. According to the rule of differentiation of the product and the tabular value of the derivative of the square root, we obtain:

You can check the solution of the problem for the derivative on derivatives calculator online .

Example 6. Find the derivative of a function

Solution. In this function, we see the quotient, the dividend of which is the square root of the independent variable. According to the rule of differentiation of the quotient, which we repeated and applied in example 4, and the table value of the derivative of the square root, we get:

To get rid of the fraction in the numerator, multiply the numerator and denominator by.

Functions complex kind it is not entirely correct to call it a "complex function". For example, it looks very impressive, but this function is not complicated, unlike.

In this article, we will deal with the concept complex function, we will learn how to identify it as part of elementary functions, give a formula for finding its derivative and consider in detail the solution of typical examples.

We will constantly use the derivative table and the rules of differentiation when solving the examples, so keep them in front of your eyes.

Complex function Is a function whose argument is also a function.

From our point of view, this definition is the most understandable. It can be conventionally denoted as f (g (x)). That is, g (x) is like an argument to the function f (g (x)).

For example, if f is the arctangent function and g (x) = lnx is the natural logarithm function, then the complex function f (g (x)) is arctan (lnx). Another example: f is the function of raising to the fourth power, and - whole rational function(look) then .

In turn, g (x) can also be a complex function. For example, ... Conventionally, such an expression can be denoted as ... Here f is the sine function, is the square root function, - fractional rational function. It is logical to assume that the degree of nesting of functions can be any finite natural number.

You can often hear that a complex function is called composition of functions.

Formula for finding the derivative of a complex function.

Example.

Find the derivative of a complex function.

Solution.

V this example f is the squaring function and g (x) = 2x + 1 is a linear function.

Here is a detailed solution using a compound function derivative formula:

Let's find this derivative after simplifying the form of the original function.

Hence,

As you can see, the results are the same.

Try not to confuse which function is f and which is g (x).

Let us explain this with an example of attentiveness.

Example.

Find derivatives of complex functions and.

Solution.

In the first case, f is the squaring function and g (x) is the sine function, so
.

In the second case, f is the sine function, and - power function... Therefore, by the formula for the product of a complex function, we have

The derivative formula for the function has the form

Example.

Differentiate function .

Solution.

In this example, a complex function can be conditionally written as , where is the sine function, the function of raising to the third power, the function of taking the logarithm to the base e, the function of taking the arctangent and the linear function, respectively.

By the formula for the derivative of a complex function

Now we find

Putting together the obtained intermediate results:

There is nothing scary, disassemble complex functions like nesting dolls.

This could be the end of the article, if not a single but ...

It is advisable to clearly understand when to apply the rules of differentiation and the table of derivatives, and when the formula for the derivative of a complex function.

NOW BE ESPECIALLY CAREFUL. We will talk about the difference between complex functions and complex functions. How much you see this difference will determine the success of finding derivatives.

Let's start with some simple examples. Function can be viewed as complex: g (x) = tgx, ... Therefore, you can immediately apply the formula for the derivative of a complex function

And here is the function difficult already can not be called.

This function is the sum of three functions, 3tgx and 1. Although - is a complex function: is a power function (quadratic parabola), and f is a tangent function. Therefore, first we apply the formula for differentiating the sum:

It remains to find the derivative of a complex function:

That's why .

We hope you get the gist.

More broadly, it can be argued that functions of a complex kind can be part of complex functions and complex functions can be part of functions of a complex kind.

As an example, let's analyze component parts function .

At first, this is a complex function that can be represented as, where f is the logarithm function to base 3, and g (x) is the sum of two functions and ... That is, .

Secondly, let us deal with the function h (x). It represents a relationship to .

This is the sum of the two functions and , where - a complex function with a numerical coefficient of 3. - cubing function, - cosine function, - linear function.

This is the sum of two functions and, where - complex function, - exponentiation function, - power function.

Thus, .

Thirdly, go to, which is the product of a complex function and a whole rational function

The function of squaring is the function of taking the logarithm to the base e.

Hence, .

Let's summarize:

Now the structure of the function is clear and it became clear which formulas and in what sequence to apply when differentiating it.

In the section on differentiating a function (finding the derivative), you can familiarize yourself with the solution of similar problems.

Decide physical tasks or examples in mathematics is completely impossible without knowledge of the derivative and methods of calculating it. The derivative is one of essential concepts mathematical analysis... We decided to devote today's article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?

Geometric and physical meaning of the derivative

Let there be a function f (x) given in some interval (a, b) ... Points х and х0 belong to this interval. When x changes, the function itself changes. Changing an argument - the difference between its values x-x0 ... This difference is written as delta x and is called argument increment. A change or increment of a function is the difference in the values ​​of a function at two points. Derivative definition:

The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.

Otherwise, it can be written like this:

What's the point in finding such a limit? And here's what:

the derivative of the function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at this point.

Physical sense derivative: the derivative of the path with respect to time is equal to the speed of the rectilinear motion.

Indeed, since school times, everyone knows that speed is a private path. x = f (t) and time t ... Average speed over a period of time:

To find out the speed of movement at a time t0 you need to calculate the limit:

Rule one: take out a constant

The constant can be moved outside the sign of the derivative. Moreover, it must be done. When solving examples in math, take as a rule - if you can simplify the expression, be sure to simplify .

Example. Let's calculate the derivative:

Rule two: derivative of the sum of functions

The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.

We will not give a proof of this theorem, but rather consider a practical example.

Find the derivative of a function:

Rule three: derivative of the product of functions

The derivative of the product of two differentiable functions is calculated by the formula:

Example: find the derivative of a function:

Solution:

It is important to say here about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.

In the above example, we meet the expression:

In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the immediate intermediate argument with respect to the independent variable.

Rule four: the quotient derivative of two functions

Formula for determining the derivative of the quotient of two functions:

We tried to tell you about derivatives for dummies from scratch. This topic is not as simple as it sounds, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.

For any question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult test and deal with tasks, even if you have never done calculating derivatives before.

After preliminary artillery preparation, examples with 3-4-5 function attachments will be less scary. Perhaps the following two examples will seem difficult to some, but if you understand them (someone will suffer), then almost everything else in the differential calculus will seem like a childish joke.

Example 2

Find the derivative of a function

As already noted, when finding the derivative of a complex function, first of all, it is necessary right UNDERSTAND the attachments. In cases where there are doubts, I recall a useful technique: we take the experimental value of "X", for example, and try (mentally or on a draft) to substitute this value in the "terrible expression".

1) First, we need to calculate the expression, which means that the amount is the deepest investment.

2) Then you need to calculate the logarithm:

4) Then raise the cosine to a cube:

5) At the fifth step, the difference:

6) And finally, the outermost function is the square root:

Complex function differentiation formula applied in reverse order, from the outermost function to the innermost. We decide:

It seems without errors:

1) Take the derivative of the square root.

2) We take the derivative of the difference using the rule

3) The derivative of the triple is zero. In the second term, we take the derivative of the degree (cube).

4) We take the derivative of the cosine.

6) And finally, we take the derivative of the deepest nesting.

It may sound too difficult, but this is not yet the most brutal example. Take, for example, Kuznetsov's collection and you will appreciate all the charm and simplicity of the analyzed derivative. I noticed that they like to give a similar thing on the exam to check whether the student understands how to find the derivative of a complex function, or does not understand.

Next example for independent decision.

Example 3

Find the derivative of a function

Hint: First, apply the linearity rules and the product differentiation rule

Complete solution and answer at the end of the tutorial.

Now is the time to move on to something more compact and cute.
It is not uncommon for an example to give a product of not two, but three functions. How to find the derivative of the product of three factors?

Example 4

Find the derivative of a function

First, let's see if it is possible to turn the product of three functions into the product of two functions? For example, if we had two polynomials in the product, then we could expand the brackets. But in this example, all functions are different: degree, exponent and logarithm.

In such cases, it is necessary consistently apply product differentiation rule twice

The trick is that for "y" we denote the product of two functions:, and for "ve" - ​​the logarithm:. Why can this be done? Is it - this is not a product of two factors and the rule does not work ?! There is nothing complicated:

Now it remains for the second time to apply the rule to the parenthesis:

You can still be perverted and put something outside the brackets, but in this case it is better to leave the answer in this form - it will be easier to check.

The considered example can be solved in the second way:

Both solutions are absolutely equivalent.

Example 5

Find the derivative of a function

This is an example for an independent solution, in the sample it is solved in the first way.

Let's look at similar examples with fractions.

Example 6

Find the derivative of a function

There are several ways to go here:

Or like this:

But the solution will be written more compactly if, first of all, we use the rule for differentiating the quotient , taking for the entire numerator:

In principle, the example is solved, and if you leave it as it is, it will not be an error. But if you have time, it is always advisable to check on a draft, but is it possible to simplify the answer?

Let us bring the expression of the numerator to common denominator and get rid of the three-story fraction:

The disadvantage of additional simplifications is that there is a risk of making a mistake not in finding the derivative, but in the case of banal school transformations. On the other hand, teachers often reject the assignment and ask to "bring to mind" the derivative.

A simpler example for a do-it-yourself solution:

Example 7

Find the derivative of a function

We continue to master the methods of finding the derivative, and now we will consider a typical case when the “terrible” logarithm is proposed for differentiation

If we follow the definition, then the derivative of a function at a point is the limit of the ratio of the increment of the function Δ y to the increment of the argument Δ x:

Everything seems to be clear. But try to calculate using this formula, say, the derivative of a function f(x) = x 2 + (2x+ 3) e x Sin x... If you do everything by definition, then after a couple of pages of calculations you will just fall asleep. Therefore, there are simpler and more effective ways.

To begin with, we note that the so-called elementary functions can be distinguished from the whole variety of functions. These are relatively simple expressions, the derivatives of which have long been calculated and entered into the table. Such functions are easy enough to remember - along with their derivatives.

Derivatives of elementary functions

Elementary functions are everything listed below. The derivatives of these functions must be known by heart. Moreover, memorizing them is not difficult at all - that's why they are elementary.

So, the derivatives of elementary functions:

Name	Function	Derivative
Constant	f(x) = C, C ∈ R	0 (yes, zero!)
Rational grade	f(x) = x n	n · x n − 1
Sinus	f(x) = sin x	cos x
Cosine	f(x) = cos x	- sin x(minus sine)
Tangent	f(x) = tg x	1 / cos 2 x
Cotangent	f(x) = ctg x	- 1 / sin 2 x
Natural logarithm	f(x) = ln x	1/x
Arbitrary logarithm	f(x) = log a x	1/(x Ln a)
Exponential function	f(x) = e x	e x(nothing changed)

If the elementary function is multiplied by an arbitrary constant, then the derivative of the new function is also easily calculated:

(C · f)’ = C · f ’.

In general, constants can be moved outside the sign of the derivative. For example:

(2x 3) ’= 2 · ( x 3) '= 2 3 x 2 = 6x 2 .

Obviously, elementary functions can be added to each other, multiplied, divided - and much more. Thus, new functions will appear, which are no longer particularly elementary, but also differentiable according to certain rules. These rules are discussed below.

Derivative of the sum and difference

Let functions f(x) and g(x), the derivatives of which are known to us. For example, you can take the elementary functions discussed above. Then you can find the derivative of the sum and difference of these functions:

1. (f + g)’ = f ’ + g ’
2. (f − g)’ = f ’ − g ’

So, the derivative of the sum (difference) of two functions is equal to the sum (difference) of the derivatives. There may be more terms. For example, ( f + g + h)’ = f ’ + g ’ + h ’.

Strictly speaking, there is no concept of "subtraction" in algebra. There is a concept of "negative element". Therefore the difference f − g can be rewritten as sum f+ (−1) g, and then only one formula remains - the derivative of the sum.

f(x) = x 2 + sin x; g(x) = x 4 + 2x 2 − 3.

Function f(x) Is the sum of two elementary functions, therefore:

f ’(x) = (x 2 + sin x)’ = (x 2) ’+ (sin x)’ = 2x+ cos x;

We reason similarly for the function g(x). Only there are already three terms (from the point of view of algebra):

g ’(x) = (x 4 + 2x 2 − 3)’ = (x 4 + 2x 2 + (−3))’ = (x 4)’ + (2x 2)’ + (−3)’ = 4x 3 + 4x + 0 = 4x · ( x 2 + 1).

Answer:
f ’(x) = 2x+ cos x;
g ’(x) = 4x · ( x 2 + 1).

Derivative of a work

Mathematics is a logical science, so many believe that if the derivative of the sum is equal to the sum of the derivatives, then the derivative of the product strike"> is equal to the product of derivatives. But figs you! The derivative of the product is calculated using a completely different formula. Namely:

(f · g) ’ = f ’ · g + f · g ’

The formula is simple, but often overlooked. And not only schoolchildren, but also students. The result is incorrectly solved problems.

Task. Find derivatives of functions: f(x) = x 3 cos x; g(x) = (x 2 + 7x- 7) e x .

Function f(x) is the product of two elementary functions, so everything is simple:

f ’(x) = (x 3 cos x)’ = (x 3) ’cos x + x 3 (cos x)’ = 3x 2 cos x + x 3 (- sin x) = x 2 (3cos x − x Sin x)

The function g(x) the first factor is a little more complicated, but general scheme does not change from this. Obviously, the first factor of the function g(x) is a polynomial, and its derivative is the derivative of the sum. We have:

g ’(x) = ((x 2 + 7x- 7) e x)’ = (x 2 + 7x- 7) ’ e x + (x 2 + 7x- 7) ( e x)’ = (2x+ 7) e x + (x 2 + 7x- 7) e x = e x· (2 x + 7 + x 2 + 7x −7) = (x 2 + 9x) · e x = x(x+ 9) e x .

Answer:
f ’(x) = x 2 (3cos x − x Sin x);
g ’(x) = x(x+ 9) e x .

Note that in the last step, the derivative is factorized. Formally, you do not need to do this, however, most derivatives are not calculated by themselves, but in order to investigate the function. This means that further the derivative will be equated to zero, its signs will be clarified, and so on. For such a case, it is better to have a factorized expression.

If there are two functions f(x) and g(x), and g(x) ≠ 0 on the set of interest to us, we can define a new function h(x) = f(x)/g(x). For such a function, you can also find a derivative:

Not weak, huh? Where did the minus come from? Why g 2? That's how! This is one of the most difficult formulas - you can't figure it out without a bottle. Therefore, it is better to study it with specific examples.

Task. Find derivatives of functions:

The numerator and denominator of each fraction contains elementary functions, so all we need is the formula for the derivative of the quotient:

By tradition, factoring the numerator into factors will greatly simplify the answer:

A complex function is not necessarily a half-kilometer long formula. For example, it is enough to take the function f(x) = sin x and replace the variable x let's say on x 2 + ln x... It will turn out f(x) = sin ( x 2 + ln x) Is a complex function. It also has a derivative, but it will not work to find it according to the rules discussed above.

How to be? In such cases, variable replacement and the formula for the derivative of a complex function help:

f ’(x) = f ’(t) · t', if x is replaced by t(x).

As a rule, with the understanding of this formula, the situation is even more sad than with the derivative of the quotient. Therefore, it is also better to explain it with specific examples, with detailed description every step.

Task. Find derivatives of functions: f(x) = e 2x + 3 ; g(x) = sin ( x 2 + ln x)

Note that if in the function f(x) instead of the expression 2 x+ 3 will be easy x then it will turn out elementary function f(x) = e x... Therefore, we make a substitution: let 2 x + 3 = t, f(x) = f(t) = e t... We are looking for the derivative of a complex function by the formula:

f ’(x) = f ’(t) · t ’ = (e t)’ · t ’ = e t · t ’

And now - attention! We carry out the reverse replacement: t = 2x+ 3. We get:

f ’(x) = e t · t ’ = e 2x+ 3 (2 x + 3)’ = e 2x+ 3 2 = 2 e 2x + 3

Now let's deal with the function g(x). Obviously, you need to replace x 2 + ln x = t... We have:

g ’(x) = g ’(t) · t’= (Sin t)’ · t’= Cos t · t ’

Reverse replacement: t = x 2 + ln x... Then:

g ’(x) = cos ( x 2 + ln x) · ( x 2 + ln x) ’= Cos ( x 2 + ln x) (2 x + 1/x).

That's all! As you can see from the last expression, the whole problem was reduced to calculating the derived sum.

Answer:
f ’(x) = 2 e 2x + 3 ;
g ’(x) = (2x + 1/x) Cos ( x 2 + ln x).

Very often in my lessons I use the word “stroke” instead of the term “derivative”. For example, a prime from the amount is equal to the sum strokes. Is that clearer? Well, that's good.

Thus, calculating the derivative comes down to getting rid of these very strokes according to the rules discussed above. As a final example, let's return to the derivative of the exponent with the rational exponent:

(x n)’ = n · x n − 1

Few know what the role n may well be a fractional number. For example, the root is x 0.5. But what if there is something fancy at the root? Again, you get a complex function - such constructions like to give on control works and exams.

Task. Find the derivative of a function:

First, let's rewrite the root as a power with a rational exponent:

f(x) = (x 2 + 8x − 7) 0,5 .

Now we make a replacement: let x 2 + 8x − 7 = t... We find the derivative by the formula:

f ’(x) = f ’(t) · t ’ = (t 0.5) ' t'= 0.5 t−0.5 t ’.

We do the reverse replacement: t = x 2 + 8x- 7. We have:

f ’(x) = 0.5 ( x 2 + 8x- 7) −0.5 x 2 + 8x- 7) ’= 0.5 · (2 x+ 8) ( x 2 + 8x − 7) −0,5 .

Finally, back to the roots: