Calculus Test -...

Evaluate \(\frac{\mathrm{d}}{\mathrm{dx}}\left(\cot ^{-1} \frac{1}{\mathrm{x}}\right)\).

If \(f(2)=4\) and \(f^{\prime}(2)=1\), then find \(\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}\).

If \(f(x)=4 x^{3}-3 x^{2}+5\), then find \(f''(x)\).

Find \(\frac{\mathrm{d}^{2} (\cot ^{-1} x)}{\mathrm{dx}^{2}}\).

Let \({f}({x})={x}-\frac{1}{{x}}\), then \({f}'(-1)\) is:

If \(y=x^{x}\), what is \(\frac{d y}{d x}\) at \(x=1\) equal to?

If \(f(x)=\left\{\begin{array}{cl}1, & \text { if } x \leq 3 \\ a x+b, & \text { if } 3

Determine the values of \(a\) and \(b\) so that \(f(x)\) is continuous.

Let \({f}({x})=2 {x}^{2}+\frac{1}{{x}}\), then \({f'}(1)\) is:

If \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), then \(\frac{d y}{d x}=?\)

If \(x^{e}=e^{x^{2}+y^{2}}\), then find \(\frac{d y}{d x}\).