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

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

If $y = \log_{2026}(\log_{2025} x)$, then $\frac{dy}{dx} = \dots \dots \dots$

Differentiate with respect to $x$ the following function: $\log _{7}(\log x)$

If $\phi(x) = \log_{5} \log_{3} x$,then $\phi'(e)$ is equal to