# NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability

NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability** **are drafted, keeping in mind your endeavour to perform well in CBSE Class 12 board exams. This chapter introduces you to the essential concepts of continuity, differentiability and relations between them. You also learn the differentiation of inverse trigonometric functions, exponential and logarithmic functions leading to techniques of differentiation.

A wide range of questions is given at the end of each topic of NCERT Class 12 Maths Chapter 5, which lets you gain in-depth knowledge of the chapter. The different types of questions asked are direct, skill-based questions, and formula-based questions. They help you practice different kinds of problem sums and thoroughly prepare for your board exams.

Our comprehensive, precise and hundred percent accurate** **NCERT Solutions for Class 12 Maths Chapter 5-Continuity and Differentiability** **cover all concepts of the chapter in detail. They help you build a sound conceptual foundation and resolve all your conceptual doubts quickly. The solutions given provide an ample amount of practice.

## NCERT Class 12 Maths Chapter 5

**Continuity**

This part of** **CBSE NCERT Solutions for Class 12 Maths Chapter 5 tells you that a function is continuous at *x* = *c* if the function is defined at *x* = *c* and if the value of the function at *x* = *c* equals the limit of the function at *x* = *c*. In case *f* is not continuous at *c*, we say f is discontinuous at *c* and *c* is called a point of discontinuity of *f*.

**Algebra of continuous functions**

This part of the Class 12 Maths Continuity and Differentiability teaches you theorem 1 which says that – if the two real functions say f and g, are continuous at a real number c, then

(i) *f* + *g* is continuous at *x* = *c*

(ii) *f* – *g* is continuous at *x *= *c*

(iii) *f ×* *g* is continuous at *x* = *c*

(iv) *f*/*g* is continuous at *x* = *c*

(provided *g*(*c*) ≠ 0)

Here theorem 2 states that suppose *f* and *g* are real-valued functions such that (*f* o *g*) is defined at *c*. If *g* is continuous at *c* and if *f* is continuous at *g* (*c*), then (*f* o *g*) is continuous at *c*.

**Differentiability**

This part of the NCERT CBSE Class 12 Maths Chapter 5 tells you that if a function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b].

Further, you learn about theorem 3 which says that if a function f is differentiable at a point c, then it is also continuous at that point.

**Derivatives of composite functions**

This portion of the chapter teaches you theorem 4, also known as chain rule theorem, this is stated below.

Let f be a real-valued function which is a composite of two functions u and v; i.e., f = v o u. Suppose t = u(x) and if both dt/dx and dv/dt exist, we have df/dx = (dv/dt). (dt/dx)

**Derivatives of implicit functions**

In this section, you learn to differentiate implicit functions.

**Derivatives of inverse trigonometric functions**

In this section, you come to know that inverse trigonometric functions are continuous functions, and chain rule is used to find derivatives of these functions.

**Exponential and Logarithmic Functions**

In this part of the chapter, you learn that the exponential function with positive base *b* > 1 is the function *y* = *f*(*x*) = *b**x* . Here you will also learn the 5 salient features of the exponential functions. Further in this section will be taught about logarithmic functions which are stated below.

Let *b* > 1 be a real number, then we say the logarithm of a to base *b* is *x* if *bx* = *a*.

The logarithm of *a* to base *b* is denoted by logb *a*. Thus, log*b a* = *x* if b*x* = a.

This part of the chapter also tells you some important properties and observations about the logarithm functions.

**Logarithmic Differentiation**

This part of the chapter tells you that a special Class of functions can be differentiated by taking their logarithm (to base e), and this process of differentiation is known as logarithmic differentiation.

**Derivatives of Functions in Parametric Forms**

This part of the chapter tells you that sometimes a third variable, called a parameter, establishes a relation between two variables that relation is expressed between two variables *x* and *y* in the form *x* = *f*(*t*), *y* = *g* (*t*) are said to be a parametric form with t as a parameter. The derivatives of such functions are found by the chain rule.

**Second-Order Derivative**

This part of the chapter teaches you that if *f *′(*x*) is differentiable, we may differentiate *dy*/*dx* = *f *′(*x*) again *w.r.t.* *x*. Then, the left-hand side becomes *d/dx (dy/dx*) which is called the second-order derivative of *y* *w.r.t. x* and is denoted by d2*y*/d*x*2.

**Mean Value Theorem**

In this section, you get to know about two fundamental results in Calculus. They have been stated below.

let *f* : [*a*, *b*] → R be continuous on [*a*, *b*] and differentiable on (*a*, *b*), such that *f(a*) = *f(b*), where *a* and *b* are some real numbers, then there exists some *c* in (*a, b*) such that *f′(c*) = 0 (Rolle’s Theorem) or *f′(c*) = [*f(b*) – *f(a*)] / (*b – a) (*Mean Value Theorem).

### NCERT Solutions for Class 12th Maths Chapter 5 – Continuity and Differentiability Exercises

CBSE NCERT Class 12 Maths Chapter 5 comprises 8 exercises, including miscellaneous exercises as well. Our NCERT Solutions Class 12 Maths Chapter 5- Continuity and Differentiability accurately answers all questions of all exercises. Our expert Maths teachers have used the best solving methods to solve all problem sums. The exercises have been discussed below.

**Exercise 5.1**

This exercise has 34 questions. Question 1 to 5, 13 to 18, 20 to 22 and then 24 to 33 ask you to prove/discuss the continuity of given functions. Questions 6 to 12, 19, 23 and 34 ask you to find points of discontinuity in functions.

**Exercise 5.2**

This exercise has 10 questions. Questions 1 to 8 is based on the differentiation of trigonometric functions. Questions 9 and 10 ask you to prove the condition given in the question.

**Exercise 5.3**

All the 15 questions of this exercise have been asked to solve for *dy/dx* of the given functions.

**Exercise 5.4**

All the 10 questions of this exercise ask you to differentiate the given functions w.r.t. *x*.

**Exercise 5.5**

There are a total of 18 questions in this exercise. Questions 1 to 11 asks to differentiate the given trigonometric functions *w.r.t.* *x*. Questions 12 to 15 are direct questions asking you to solve for *dy/dx*. Questions 16 to 18 ask to find derivatives and differentiate the given functions.

**Exercise 5.6**

There are a total of 11 questions in this exercise based on finding derivatives of parametric functions.

**Exercise 5.7**

There are a total of 17 direct and application-based questions in this exercise based on second-order derivatives.

**Miscellaneous Exercise**

There are a total of 23 questions in this exercise based on the concepts learnt in this chapter.

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