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 $y=\log \sqrt{\frac{1+\sin x}{1-\sin x}}$,then $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$ is

If $y = \log \log x$,then $e^y \frac{dy}{dx} = $

$\frac{d}{dx}(e^{x \log x} + e^3) = $ . . . . . .

If $y = \log_{10}x + \log_x 10 + \log_x x + \log_{10} 10$,then $\frac{dy}{dx} = $