#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

## Chapter 10: Differentiability

#### Chapter 10: Differentiability Exercise 10.1 solutions [Pages 10 - 11]

Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.

Show that f(x) = x^{1}^{/3} is not differentiable at x = 0.

Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at *x* = 3. Also, find f'(3).

Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:

Discuss the continuity and differentiability of the

Find whether the function is differentiable at *x* = 1 and *x* = 2

Show that the function

(i) differentiable at x = 0, if m > 1

(ii) continuous but not differentiable at *x* = 0, if 0 < m < 1

(iii) neither continuous nor differentiable, if m ≤ 0

Find the values of *a* and *b* so that the function

Show that the function

\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at *x* = 1.

If \[f\left( x \right) = \begin{cases}a x^2 - b, & \text { if }\left| x \right| < 1 \\ \frac{1}{\left| x \right|} , & \text { if }\left| x \right| \geq 1\end{cases}\] is differentiable at *x* = 1, find a, b.

Find the values of *a* and *b*, if the function *f* defined by

*x*= 1.

#### Chapter 10: Differentiability Exercise 10.2 solutions [Page 16]

If f is defined by f (x) = x^{2}, find f'(2).

If *f *is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]

Show that the derivative of the function *f* given by

If for the function

\[\Phi \left( x \right) = \lambda x^2 + 7x - 4, \Phi'\left( 5 \right) = 97, \text { find } \lambda .\]

If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]

, find f'(4).

Find the derivative of the function f defined by f (x) = mx + c at x = 0.

Examine the differentialibilty of the function* f *defined by

\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]

Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.

Discuss the continuity and differentiability of f (x) = |log |x||.

Discuss the continuity and differentiability of f (x) = e^{|x| }.

Discuss the continuity and differentiability of

Is |sin *x*| differentiable? What about cos |*x*|?

#### Chapter 10: Differentiability solutions [Page 17]

Define differentiability of a function at a point.

Is every differentiable function continuous?

Is every continuous function differentiable?

Give an example of a function which is continuos but not differentiable at at a point.

If f (x) is differentiable at x = c, then write the value of

If f (x) = |x − 2| write whether f' (2) exists or not.

Write the points where f (x) = |log_{e} x| is not differentiable.

Write the points of non-differentiability of

Write the derivative of f (x) = |x|^{3} at x = 0.

Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.

If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of

#### Chapter 10: Differentiability solutions [Pages 17 - 20]

Let f (x) = |x| and g (x) = |x^{3}|, then

f (x) and g (x) both are continuous at x = 0

f (x) and g (x) both are differentiable at x = 0

f (x) is differentiable but g (x) is not differentiable at x = 0

f (x) and g (x) both are not differentiable at x = 0

The function *f* *(x*) = sin^{−1} (cos *x*) is

discontinuous at

*x*= 0continuous at

*x*= 0differentiable at

*x*= 0none of these

The set of points where the function *f* (*x*) = *x* |*x*| is differentiable is

\[\left( - \infty , \infty \right)\]

\[\left( - \infty , 0 \right) \cup \left( 0, \infty \right)\]

\[\left( 0, \infty \right)\]

\[\left[ 0, \infty \right]\]

If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is

continuous at x = − 2

not continuous at x = − 2

differentiable at x = − 2

continuous but not derivable at x = − 2

Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]

f is continuous

f is differentiable for some x

f' is continuous

f'' is continuous

The function f (x) = e^{−}^{|x|} is

continuous everywhere but not differentiable at x = 0

continuous and differentiable everywhere

not continuous at x = 0

none of these

The function f (x) = |cos x| is

everywhere continuous and differentiable

everywhere continuous but not differentiable at (2n + 1) π/2, n ∈ Z

neither continuous nor differentiable at (2n + 1) π/2, n ∈ Z

none of these

If \[f\left( x \right) = \sqrt{1 - \sqrt{1 - x^2}},\text{ then } f \left( x \right)\text { is }\]

continuous on [−1, 1] and differentiable on (−1, 1)

continuous on [−1, 1] and differentiable on

\[\left( - 1, 0 \right) \cup \left( 0, 1 \right)\]continuous and differentiable on [−1, 1]

none of these

If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]

\[a = b = c = 0\]

\[a = 0, b = 0; c \in R\]

\[b = c = 0, a \in R\]

\[c = 0, a = 0, b \in R\]

If \[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . ,\]

then at x = 0, f (x)

has no limit

is discontinuous

is continuous but not differentiable

is differentiable

If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]

\[f' \left( 1^+ \right) = 1\]

\[f' \left( 1 \right) = - 1\]

\[f' \left( 1 \right) = 1\]

\[f' \left( 1 \right) = - 1\]

If \[f\left( x \right) = \left| \log_e |x| \right|\]

f (x) is continuous and differentiable for all x in its domain

f (x) is continuous for all for all × in its domain but not differentiable at x = ± 1

(x) is neither continuous nor differentiable at x = ± 1

none of these

Let \[f\left( x \right) = \begin{cases}\frac{1}{\left| x \right|} & for \left| x \right| \geq 1 \\ a x^2 + b & for \left| x \right| < 1\end{cases}\] If *f* (*x*) is continuous and differentiable at any point, then

\[a = \frac{1}{2}, b = - \frac{3}{2}\]

\[a = - \frac{1}{2}, b = \frac{3}{2}\]

a = 1, b = − 1

none of these

The function f (x) = x − [x], where [⋅] denotes the greatest integer function is

continuous everywhere

continuous at integer points only

continuous at non-integer points only

differentiable everywhere

Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if

a = 2

a = 1

a = 0

a = 1/2

Let f (x) = |sin x|. Then,

f (x) is everywhere differentiable.

f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z

f (x) is everywhere continuous but not differentiable at \[x = \left( 2n + 1 \right)\frac{\pi}{2}, n \in Z\]

none of these

Let *f* (*x*) = |cos *x*|. Then,

^{f (x) is everywhere differentable}f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z

f (x) is everywhere continuous but not differentiable at \[x = \left( 2n + 1 \right)\frac{\pi}{2}, n \in Z\].

(d) none of these

The function f (x) = 1 + |cos x| is

continuous no where

continuous everywhere

not differentiable at x = 0

not differentiable at x = n π, n ∈ Z

The function f (x) = |cos x| is

differentiable at x = (2n + 1) π/2, n ∈ Z

continuous but not differentiable at x = (2n + 1) π/2, n ∈ Z

neither differentiable nor continuous at x = n ∈ Z

none of these

The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is

continuous as well as differentiable for all

*x*∈ Rcontinuous for all

*x*but not differentiable at some xdifferentiable for all x but not continuous at some x.

none of these

Let f (x) = a + b |x| + c |x|^{4}, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if

a = 0

b = 0

c = 0

none of these

If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is

continuous and differentiable at x = 3

continuous but not differentiable at x = 3

differentiable nut not continuous at x = 3

neither differentiable nor continuous at x = 3

If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\] then f (x) is

continuous as well as differentiable at x = 0

continuous but not differentiable at x = 0

differentiable but not continuous at x = 0

none of these

If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\]

then at x = 0, f (x) is

continuous and differentiable

differentiable but not continuous

continuous but not differentiable

neither continuous nor differentiable

The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is

R

R − {3}

(0, ∞)

none of these

Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is

continuous at x = − 1

differentiable at x = − 1

everywhere continuous

everywhere differentiable

## Chapter 10: Differentiability

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

#### Textbook solutions for Class 12

## RD Sharma solutions for Class 12 Mathematics chapter 10 - Differentiability

RD Sharma solutions for Class 12 Maths chapter 10 (Differentiability) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 10 Differentiability are Higher Order Derivative, Algebra of Continuous Functions, Derivative - Exponential and Log, Concept of Differentiability, Proof Derivative X^n Sin Cos Tan, Infinite Series, Continuous Function of Point, Mean Value Theorem, Second Order Derivative, Derivatives of Functions in Parametric Forms, Logarithmic Differentiation, Exponential and Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Inverse Trigonometric Functions, Derivatives of Composite Functions - Chain Rule, Concept of Continuity.

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