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

Chapter 10: Differentiability
RD Sharma solutions for Class 12 Maths Chapter 10 Differentiability Exercise 10.1 [Pages 10 - 11]
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Show that f(x) = x1/3 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
RD Sharma solutions for Class 12 Maths Chapter 10 Differentiability Exercise 10.2 [Page 16]
If f is defined by f (x) = x2, 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|?
RD Sharma solutions for Class 12 Maths Chapter 10 Differentiability [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) = |loge 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
RD Sharma solutions for Class 12 Maths Chapter 10 Differentiability [Pages 17 - 20]
Let f (x) = |x| and g (x) = |x3|, 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 = 0
continuous at x = 0
differentiable at x = 0
none 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 ∈ R
continuous for all x but not differentiable at some x
differentiable 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 solutions for Class 12 Maths 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 Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.
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Concepts covered in Class 12 Maths 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.
Using RD Sharma Class 12 solutions Differentiability exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.
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