Application of the derivative for proving identities, inequalities and solving equations

Aim: Broadening students' knowledge about the potential of first and second derivative of a function

Application: Proving some types of identities and inequalities as well as solving equations

Presentation structure:
Proving identities using a derivative

• algebra identities
• problems for individual work
• trigonometric problems
• problems for individual work

# 1  Proving identities using derivatives

###### The basic theorem

Theorem 1.1. If a function is defined and differentiable in the interval and for every , then is a constant in .

Corollary 1.2. If differentiable in the interval functions and have equal derivatives everywhere in , then they differ with a constant, i.e. if , for every , then , - a constant.

We take on trust the truthfulness of the theorem above.

###### Identity

Definition 1.3. Two expressions and (rational or irrational) are called identical in a given set , in which and are defined if for all values of variables from their numerical values are equal. Then the equality is called identity. Usually for is used the set of admissible values of the equality .

There are several ways to prove that the equality is identical with the given set :

1. If the expression is more complex it is transformed into an identical expression (in the set ), after that - in an identical to it and so on until after a finite number of transformations ( for example ) the expression is identical to , i.e. . This leads to the conclusion
2. The difference is transformed into an identical to it expression , then into an identical to it expression and so on until it is proven that is identically equal to zero, i.e. . From here , i.e. .
3. The expressions and are simultaneously transformed into respective identical and . Further on and into and and so on until and become apparently equal i.e. .

Of bigger interest to us is the second method. There are cases when using traditional means (from elementary mathematics) it is hard or impossible to directly prove that . But it is possible (by means of a special technique) to show first that , and then that this is equal to zero.

###### Algorithm

When proving identities of the kind we implement the following algorithm:

1. We consider the functionwhere are parameters.
2. We prove that for every admissible value of .
3. We determine that where is a randomly chosen number from the admissible values of .
4. On the basis of theorem 1.1 we conclude that for every admissible . Consequently.

## 1.1  Proving algebra identities

Example 1.4. Prove the identityfor every xq a is a constant.

Solution: Let .

. When ,

.

Example 1.5. Prove the identity for every x, y.

Solution: Let .

.

When . Consequently .

Example 1.6. Prove the identity for every x, y, z.

Solution:

It is apparent that for most of these problems using derivatives leads to lowering of exponents of the expressions and so eases calculations.
The basic theorem can be used for proving the following identities.

Example 1.7.  , if

This problem represents a conditional identity which is a corollary of Example 1.6. In this case we offer a different solution.

Solution: From . Considering the function

Consequently . When if .

Example 1.8. Prove the identity for every a, b, c.

Solution: Let

Consequently . When .

Example 1.9.

Prove the identity

Hint: Let the function considered be .

Example 1.10. Prove the identity for every a, b, c.

Solution:

When

Example 1.11. Prove the identity for every a, b, c.

Solution: Let

.

When .

Consequently .

Example 1.12. Prove the identity .

Hint: Let .

Consequently . When

#### For individual work

Exercise 1.13. Prove the identity for every x, y.

Exercise 1.14. Prove the identity for every x,y.

Exercise 1.15. Prove the identity for every x, y, z.

Exercise 1.16. Prove the identity for every x, y, z.

Exercise 1.17. Prove the identity for every a, b, x, y, z.

Exercise 1.18. Prove the identity for every a, b, c.

Exercise 1.19. Prove the identity for every a, b, c.

# 2  Proving trigonometric identities

The algorithm offered can be used for proving trigonometric identities. It has to be mentioned, though, that the derivatives of differences which result from subtraction of trigonometric expressions, rather than the identities themselves which have to be proven. This means that more complex transformations have to be carried out to prove that . This is why we will only illustrate this application with a few examples, and the rest will remain to be done individually.

Prove the identities:

Example 2.1. Prove the identity for every x.

Solution: Let

Then . When the result is that

. Consequently

Example 2.2. Prove the identity for every x.

Solution: Let

When .

Example 2.3. Prove the identity for every .

Solution:

Consequently . When .

Example 2.4. Prove the identity for every .

Solution: Let .

Consequently . When

Example 2.5. Prove the identity for every

Hint: Consider this function: . Determine that Establish the constant for

For trigonometric identities with two or more variables we do the following: we introduce the function , as a function of one of these variables. The remaining are treated as constants. In this case after finding the first derivative of , many of the summands cease to play a part since their derivatives are zeros. And though some of the problems require a lot of calculation, they are simpler and easier.

Example 2.6. Prove the identity for every x.

Solution: We assume that is a constant and consider the function

Consequently . Let . Then .

Example 2.7. Prove the identity for every .

Solution: Let's assume that is a constant. Then

When , we determine that .

Example 2.8. Prove the identity for every

Solution: .

When

The basic theorem can be used for proving conditional trigonometric identities.

Example 2.9. Prove the identity , if .

Solution: ,

When .

Example 2.10. Prove the identity , if .

Solution:

When.

Example 2.11. Prove the identity if .

Solution:

When . Consequently.

Example 2.12. Prove the identity , if , .

Hint: Consider . Find and .

Example 2.13. Prove the identity , if .

Hint: Let

Determine and find when.

#### For individual work

Prove the following identities:

Problem 2.14.  for every

Problem 2.15.  for every

Problem 2.16.

Problem 2.17.

Problem 2.18.

Problem 2.19.

Problem 2.20.

Problem 2.21.

Problem 2.22.

Problem 2.23.  , if

Problem 2.24.  , if

Problem 2.25.  , if

Problem 2.26.  , if .

By Ilia Makrelov, Plovdiv university, ilmak@uni-plovdiv.bg