RD Sharma solutions for Mathematics Volume 1 and 2 [English] Class 12 chapter 9 - Differentiability [Latest edition]

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 

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

Examine the differentialibilty of the function f defined by

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

Discuss the continuity and differentiability of 

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

Write the points of non-differentiability of 

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

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

Let f (x) = |x| and g (x) = |x³|, then

The function f (x) = sin⁻¹ (cos x) is

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

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

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

The function f (x) = e^−|x| is

The function f (x) = |cos x| is

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

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

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)

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

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

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

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

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 

Let f (x) = |sin x|. Then,

Let f (x) = |cos x|. Then,

The function f (x) = 1 + |cos x| is

The function f (x) =  |cos x| is

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

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

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

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 

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

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

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