# Differentiation of Exponential and Logarithmic Functions, Objectives, Derivative of Exponential Function, Derivative of Logarithmic Function

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## Objectives

After studying this lesson, you will be able to:

Define and find the derivatives of exponential and logarithmic functions;

Find the derivatives of functions expressed as a combination of algebraic, trigonometric, exponential and logarithmic functions; and

Find second order derivative of a function.

State Rolle’s Theorem and Lagrange’s Mean Value Theorem; and

Test the validity of the above theorems and apply them to solve problems.

## Derivative of Exponential Function:

Let be an exponential function.

(Corresponding small increments)

From (1) and (2), we have

Dividing both sides by and taking the limit as

Thus, we have .

Working rule:

Example:

Find the derivative of each of the following functions:

(i) (ii)

Solution:

(i) Let

Then where

and

We know that,

Alternatively

(ii) Let

Then where

and

We know that,

## Derivative of Logarithmic Function:

We first consider logarithmic function

Let

( and are corresponding small increments in and )

From (1) and (2), we get

[Multiply and divide by ]

Taking limits of both sides, as , we get

Thus,

Working rule:

Example:

Find the derivative of each of the functions given below:

(i) (ii)

Solution:

(i)

(ii) or