Let $f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}$ for $1 < x < 26$ be a real-valued function. Then $f'(x)$ for $1 < x < 26$ is

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$ for all real $x$ and $y$. If $\lim_{x \rightarrow 0} \frac{f(x)}{x} = 1$,then $f'(3)$ is equal to:

Let $f: (-1, 1) \to R$ be a differentiable function with $f(0) = -1$ and $f'(0) = 1$. Let $g(x) = [f(2f(x) + 2)]^2$. Then $g'(0) = $

The differential coefficient of $f(\sin x)$ with respect to $x,$ where $f(x) = \log x,$ is

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{x - 2}{x^2 - 3x + 2}, & x \in R - \{1, 2\} \\ 2, & x = 1 \\ 1, & x = 2 \end{cases}$,then $\lim_{x \rightarrow 2} \frac{f(x) - f(2)}{x - 2} = $

$\frac{d}{dx}(e^{x\sin x}) = $