# NCERT Class 12-Mathematics: Chapter –5 Continuity and Differentiability Part 2

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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

### Short Answer (S.A)

**Question 1:**

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

**Answer:**

It is given that the function is continuous at . Therefore

Thus, is continuous at if .

**Question 2:**

Discuss the continuity of the function *.*

**Answer:**

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 *.*

**Answer:**

Given

Now,

As is continuous at , we have