Continuity and ...

Consider the function \(f(x)=\left\{\begin{array}{cc}x^{2} \ln |x| & x \neq 0 \\ 0 & x=0\end{array}\right.\). What is f’(0) equal to?

If \(y+\sin ^{-1}\left(1-x^{2}\right)=e^{x}\), then \(\frac{d y}{d x}=?\) 

If \(y=e^{x+e^{x+e^{x+\cdots \infty}}}\), then \(\frac{d y}{d x}\) is:

If \(f(x)=x^{3}+3 x^{2}+3 x-7\), then find the value of \(\frac{d f(x)}{d x}\) at \(x=2\).

If \(y=3 t^{2}-4 t-3\) and \(x=8 t+5\), find \(\frac{d y}{d x}\) at \(t=6\).

If \(y=\frac{12 \tan x-4 \tan ^{3} x}{9-27 \tan ^{2} x}\), then find \(\frac{d^{2} y}{d x^{2}}\).

Find the value of \(\lim _{x \rightarrow 0} \frac{\sqrt[p]{1+x}-1}{x}\), where \(p\) is a positive integer.

What is the value of \(\frac{d y}{d x}\), if \(y^{2}+x^{2}+3 x+5=0\) at \((0,-3)?\)

Let \(f(x)=\log x^{3}+2 x^{2}-3 x+100\), then find \(f'(3)\).

What is the value of \(\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{3 x}\)?

If \(f(x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{e^{2 x}-1}, & x \neq 0 \\ k-2, & x=0\end{array}\right.\) is continuous at \(x=0\), then \(k=\)?

If \(\lim _{x \rightarrow 0} \frac{\log (1+\sin x)}{x}=k\), the value of \(k\) is:

What is the value of \(\lim _{x \rightarrow 1} \frac{4 x^{2}-5 x+1}{x^{2}-1}\)?

If \(\sqrt{y}=\sin x+\cos x\), then \(\frac{d y}{d x}\) is:

Let \(f(x)=2 x^{3}+\frac{1}{x}\), then \(f'(1)\) is:

What is \(\lim _{x \rightarrow 0} \frac{\sin x \log (1-x)}{x^{2}}\) equal to?

Find the derivative of \(\frac{1}{3 x^{2}}\) with respect to \(x\).

What is \(\lim _{x \rightarrow 0} \frac{3^{x}+3^{-x}-2}{x}\) equal to?

If \(f(x)=\frac{\sin \left(e^{x-2}-1\right)}{\log (x-1)}, x \neq 2\) and \(f(x)=k\). Then, the value of k for which f will be continuous at x = 2 is:

The derivative of \(\csc ^{-1}\left(\frac{1}{2 x^{2}-1}\right)\) with respect to \(\sqrt{1-x^{2}}\) is:

Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\), if \(\mathrm{y}=\tan ^{-1}\left[\frac{8 \mathrm{x}}{1-15 \mathrm{x}^{2}}\right]\).

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

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

Let \(f(2)=2\) and \(f(x)=2\). Then, \(\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}\) is given by:

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

Let \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}^{2}+\frac{1}{\mathrm{x}}\), then \(\mathrm{f'}(1)\) 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 \(\frac{d y}{d x}\).