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Find the derivative of $f(x) = 1 + x + x^{2} + x^{3} + \dots + x^{50}$ at $x = 1$.

If $f(1)=1$ and $f^{\prime}(1)=3$,then the derivative of $f(f(f(x)))+(f(x))^2$ at $x=1$ is

If $f(x) = e^x g(x)$,$g(0) = 2$,and $g'(0) = 1$,then $f'(0)$ is

If $f(x) = \begin{cases} x-5, & \text{for } x \leq 1 \\ 4x^2-9, & \text{for } 1 < x < 2 \\ 3x+4, & \text{for } x \geq 2 \end{cases}$,then $f^{\prime}(2^{+})$ is equal to:

If $y = \tan^{-1}\left(\frac{4x}{1+5x^2}\right) + \cot^{-1}\left(\frac{3-2x}{2+3x}\right)$,then $\frac{dy}{dx}$ is equal to

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