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

Get unlimited access to the best preparation resource for CTET-Hindi/Paper-2 Child-Development-and-Pedagogy: fully solved questions with step-by-step explanation- practice your way to success.

Download PDF of This Page (Size: 136K)

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

Developed by: