$\frac{d}{dx} \left[ \log \sqrt{\frac{1 - \cos x}{1 + \cos x}} \right] = $

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

Let $\ln x$ denote the logarithm of $x$ with respect to the base $e$. Let $S \subset R$ be the set of all points where the function $\ln(x^2-1)$ is well-defined. Then,the number of functions $f: S \rightarrow R$ that are differentiable,satisfy $f^{\prime}(x)=\ln(x^2-1)$ for all $x \in S$ and $f(2)=0$,is

If $y = 3^{x^2}$,then $\frac{dy}{dx}$ is equal to

$m$ is the slope of a tangent to the curve $e^{y}=1+x^2$ at $x=1$,then $m=$

If $f(x)=3^x$ and $g(x)=4^x$,then $\frac{f^{\prime}(0)-g^{\prime}(0)}{1+f^{\prime}(0) g^{\prime}(0)}$ is