If $f(x) = e^{g(x)}$ and $g(x) = \int_{2}^{x} \frac{t}{1 + t^4} \, dt$,then $f'(2)$ has the value equal to:

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 $y = \log_{\sin x}(\tan x)$,then $\left( \frac{dy}{dx} \right)_{\pi/4} = $

If $y = e^{(1 + \log_e x)}$,then the value of $\frac{dy}{dx} = $

Mrs. Rajni deposited $Rs. 10,000$ in a bank that pays $4 \%$ interest compounded continuously. The amount she gets after $10$ years is approximately $Rs.$ . . . . . . . (Given $e^{(0.4)} = 1.49182$)

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