# NCERT Class 12-Mathematics: Chapter – 5 Continuity and Differentiability Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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5.1. 11 Exponential and logarithmic functions

(i) The exponential function with positive base is the function . Its domain is R, the set of all real numbers and range is the set of all positive real numbers. Exponential function with base is called the common exponential function and with base e is called the natural exponential function.

(ii) Let be a real number. Then we say logarithm of to base b is if , Logarithm of a to the base b is denoted by . If the base , we say it is common logarithm and if , then we say it is natural logarithms. denotes the logarithm function to base e. The domain of logarithm function is , the set of all positive real numbers and the range is the set of all real numbers.

(iii) The properties of logarithmic function to any base are listed below:

1.

2.

3.

4.

5.

6.

(iv) The derivative of is , i.e.. , . The derivative of w. r. t. , is

5.1. 12 Logarithmic differentiation is a powerful technique to differentiate functions of the form , where both f and u need to be positive functions for this technique to make sense.

5.1. 13 Differentiation of a function with respect to another function Let and be two functions of , then to find derivative of w. r. t. to , i.e.. , to find , we use the formula

5.1. 14 Second order derivative

is called the second order derivative of w. r. t. . It is denoted by ′′ or , if .

5.1. 15 Rolle՚s Theorem

Let be continuous on and differentiable on , such that , where and are some real numbers. Then there exists at least one point in such that . Geometrically Rolle՚s theorem ensures that there is at least one point on the curve at which tangent is parallel to .

5.1. 16 Mean Value Theorem (Lagrange)

Let be a continuous function on and differentiable on . Then there exists at least one point in such that .

Geometrically, Mean Value Theorem states that there exists at least one point in such that the tangent at the point is parallel to the secant joining the points and .

## 5.2 Solved Examples

Question 1:

Find the value of the constant so that the function defined below is continuous at , where

It is given that the function is continuous at . Therefore

Thus, is continuous at if .

Question 2:

Discuss the continuity of the function .

Since sin and are continuous functions and product of two continuous function is a continuous function, therefore is a continuous function.

Question 3:

If is continuous at , find the value of .