If $f(1) = 3$ and $f'(1) = 2$,then find the value of $\frac{d}{dx} \{ \log f(e^x + 2x) \}$ at $x = 0$.

The derivative of $(\log x)^{\sin x}$ with respect to $\cos x$ at $x=\frac{\pi}{2}$ is

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 $e^{(0.4)} = 1.49182$)

If $\frac{d}{dx} \left( A \log \left( \frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1} \right) \right) = \frac{1}{x \sqrt{1-x^3}}$,then $AB=$

$\frac{d}{dx} \left( \log \sqrt{\frac{1+\sin x}{1-\sin x}} \right) = $

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