Let $f(x)=e^x$, $g(x)=\sin^{-1} x$ and $h(x)=f(g(x))$, then $\frac{h'(x)}{h(x)}$ is equal to

The derivative of $(\sin x)^x$ with respect to $x^{(\sin x)}$ is

If the slope of the tangent drawn to the curve $y=e^{a+bx^2}$ at the point $P(1,1)$ is $-2$, then the value of $2a-3b$ is

$\frac{d}{dx} \left( \log \left( \sqrt{x + \sqrt{x^2 + a^2}} \right) \right) = $

$f(x) = x^{x}$ has a stationary point at

If $y = \log^n x$, where $\log^n$ denotes the $n$-th iterated logarithm $\log_e(\log_e(\dots \log_e x \dots))$ ($n$ times), then $x \log x \log^2 x \log^3 x \dots \log^{n-1} x \log^n x \frac{dy}{dx}$ is equal to