DERIVATIVES OF VARIABLES

In this section of variables, we will extend the theorems of limits covered previously. Recall that a derivative is actually a limit. The proof of the theorems presented here involve the delta process.

POWER FORM

Theorem 2. The derivative of the function

is given by

if n is any real number. PROOF: By definition

The expression may be expanded by the binomial theorem into

Substituting in the expression for the derivative, we have

Simplifying, this becomes

Letting x approach zero, we have

Thus, the proof is complete.

EXAMPLE: Find the derivative of

SOLUTION: Apply Theorem 2, such that,

Therefore,

n=5

and

n-1=4

so that given

and substituting values for n, find that

EXAMPLE: Find the derivative of

SOLUTION Apply Theorem 2, such that,

Therefore,

and

n-1=0

so that

The previous example is a special case of the power form and indicates that the derivative of a function with respect to itself is 1.

EXAMPLE: Find the derivative of

where a is a constant. SOLUTION

and

so that

Therefore,

Table 5-1.-Derivatives of Functions

The previous example is a continuation of the derivative of a function with respect to itself and indicates that the derivative of a function with respect to itself, times a constant, is that constant.

EXAMPLE. Find the derivative of

SOLUTION:

A study of the functions and their derivatives in table 5-1 should further the understanding of this section.

PRACTICE PROBLEMS:

Find the derivatives of the following:

ANSWERS: