Find f′(x) if f (x) = (sin x)^{sin x} for all 0 < x < π.

4

(a)

Find the value of k such that the function
\(f(x)=\begin{cases} \frac { { 2 }^{ x+2 }-16| }{ { 4 }^{ x }-16 } \ , \ \ if\quad x\neq 2 \\ \quad k\quad \quad , \ \ if\quad x=2 \end{cases}\)is continuous at x=2

4

(a)

Find the derivative of each of the following function w.r.t. x, or find \(\frac { dy }{ dx } \):
\(y={ sin }^{ -1 }\left[ x\sqrt { 1-x } -\sqrt { x } \sqrt { 1-{ x }^{ 2 } } \right] \)

4

(a)

If x^py^q = (x + y)^p+q, prove that \((i)\frac { dy }{ dx } =\frac { y }{ x } and\quad (ii)\frac { { d }^{ 2 }y }{ { d }x^{ 2 } } =0.\)

4

(a)

Verify MVT for the following functions:
f (x) =\(\frac { 1 }{ 4x-1 } \) in [1,4].

4

(a)

Examine the differentiability of the function f(x)=\(\begin{cases} x\left[ x \right] \quad \ \ \ \ \ ,\quad if\quad 0\le x< \\ \left( x-1 \right) x \ \ , \quad if\quad 2\le x<3 \end{cases}\quad \)at x=2

4

(a)

Show that f (x) = |x-5| is continuous but not differentiable at x = 5.

4

(a)

Examine the continuity of a function 
\(f(x)=\left\{\begin{array}{c} |x| \cos \frac{1}{x}, \text { if } x \neq 0 \\ 0, \quad \text { if } x=0 \end{array}\right. \text { at } x=0\)

4

(a)

Show that the function \(f(x)=\left\{\begin{array}{c} \frac{e^{1 / x}-1}{e^{1 / x}+1}, \text { when } x \neq 0 \\ 0, \quad \text { when } x=0 \end{array}\right.\) is discontinuous at x = 0.

4

(a)

Let \(f(x)=x|x|, \forall x \in R\) ,Discuss the derivability of f(x) at x = 0.

4

(a)