Calculus Test -...

If \(y+\sin ^{-1}\left(1-x^{2}\right)=e^{x}\), then \(\frac{d y}{d x}=?\) 

For what choice of \(a\) and \(b\) is the function \(f(x)=\left\{\begin{array}{ll}x^2,\quad\quad x \leq c \\ a x+b, \quad x>c\end{array}\right.\) is differentiable at \(x=c\).

If \(y=e^{x+e^{x+e^{x+\cdots \infty}}}\), then \(\frac{d y}{d x}\) is:

If \(f(x)=x^{3}+3 x^{2}+3 x-7\), then find the value of \(\frac{d f(x)}{d x}\) at \(x=2\).

If \(y=3 t^{2}-4 t-3\) and \(x=8 t+5\), find \(\frac{d y}{d x}\) at \(t=6\).

If \(y=\frac{12 \tan x-4 \tan ^{3} x}{9-27 \tan ^{2} x}\), then find \(\frac{d^{2} y}{d x^{2}}\).

What is the value of \(\frac{d y}{d x}\), if \(y^{2}+x^{2}+3 x+5=0\) at \((0,-3)?\)

Let \(f(x)=\log x^{3}+2 x^{2}-3 x+100\), then find \(f'(3)\).

Find the value of the constant \(\lambda\) so that the function given below is continuous at \(x=-1\)

\(f(x)=\left\{\begin{array}{cc} \frac{x^2-2 x-3}{x+1}, & x \neq-1 \\ \end{array}\right.\)

If \(f(x)= \begin{cases}x^2+3 x+a & , \text { for } x \leq 1 \\ b x+2 & \text {, for } x>1\end{cases}\) is everywhere differentiable, find the values of \(a\) and \(b\).