For what value of a and b is the function \(f\left( x \right) =\begin{cases} { x }^{ 2 },\quad \quad x\le c \\ ax+b,\quad x>c \end{cases}\) is differentiable at x = c.

Differentiate \({ tan }^{ -1 }(secx+tanx),\) \(-\frac{\pi}{ 2 }\) with respect to 'x'.

Differentiate \({ \left( \sin { x } \right) }^{ { \cos { ^{ -1x } } } }\) with respect to 'x'.

If \(x=\tan { \left( \frac { 1 }{ a } \log { y } \right) } \) then show that \(\left( 1+{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +(2x-a)\frac { dy }{ dx } =0.\)

Discuss the differentiability of the functions:
(i) \(f(x)=\{ \begin{matrix} 1,0\le x\le 1 \\ x,x>1 \end{matrix}at=1\)
(ii) \(f(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}=\lim _{h \rightarrow \infty} \frac{1+h-1}{h}=1\)