Differentiation

If $y=\sin ^{-1}\left(\frac{3 x}{2}-\frac{x^3}{2}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

If $y=\left((x+1)(4 x+1)(9 x+1) \ldots\left(\mathrm{n}^2 x+1\right)\right)^2$, then $\frac{\mathrm{dy}}{\mathrm{d} x}$ at $x=0$ is

If the function $\mathrm{f}(x)=x^3+\mathrm{e}^{\frac{x}{2}}$ and $\mathrm{g}(x)=\mathrm{f}^{-1}(x)$ then the value of $g^{\prime}(1)$ is

If $y=A \cos \mathrm{n} x+\mathrm{B} \sin \mathrm{nx}$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=$

If $y=\sec \left(\tan ^{-1} x\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1$ is equal to

 Let $f$ be a twice differentiable function such that $\mathrm{f}^{\prime \prime}(x)=-\mathrm{f}(x), \mathrm{f}^{\prime}(x)=\mathrm{g}(x)$ and $\mathrm{h}(x)=[\mathrm{f}(x)]^2+[\mathrm{g}(x)]^2$. If $\mathrm{h}(5)=1$, then $\mathrm{h}(10)$ ...

If $y=\log \left[\mathrm{e}^{5 x}\left(\frac{3 x-4}{x+5}\right)^{\frac{4}{3}}\right]$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

Derivative of $\sin ^2 x$ with respect to $e^{\cos x}$

If $y=[(x+1)(2 x+1)(3 x+1) \ldots \ldots \ldots(n x+1)]^4$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=0$ is

If $y=a \sin x+b \cos x \quad$ (where $\mathrm{a}$ and $\mathrm{b}$ are constants), then $y^2+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2$ is

If $y=\sqrt{\frac{1-\sin ^{-1} x}{1+\sin ^{-1} x}}$, then $\left(\frac{d y}{d x}\right)$ at $x=0$ is

If $y=\left[\mathrm{e}^{4 x}\left(\frac{x-4}{x+3}\right)^{\frac{3}{4}}\right]$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}=$

If $\mathrm{F}(x)=\left(\mathrm{f}\left(\frac{x}{2}\right)\right)^2+\left(\mathrm{g}\left(\frac{x}{2}\right)\right)^2$, where $\mathrm{f}^{\prime \prime}(x)=-\mathrm{f}(x)$ and $\mathrm{g}(x)=\mathrm{f}^{\prime}(x)$ and given by $\mathrm{F}...

If $x=\sec \theta-\cos \theta, y=\sec ^{10} \theta-\cos ^{10} \theta$ and $\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2=k\left(y^2+4\right)$, then the value of $k$ is

If $\mathrm{f}(x)=\sin ^{-1}\left(\frac{2 \cdot 3^x}{1+9^x}\right)$, then $\mathrm{f}^{\prime}\left(\frac{1}{2}\right)$ equals

If $\mathrm{f}(x)=\log _{x^2}\left(\log _{\mathrm{e}} x\right)$, then $\mathrm{f}^{\prime}(x)$ at $x=\mathrm{e}$ is

If $y=\sec \left(\tan ^{-1} x\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1$ is equal to

If $f(1)=1, f^{\prime}(1)=5$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

If $\mathrm{f}(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x}$ is decreasing for all $x$ then

If $\mathrm{g}(x)=[\mathrm{f}(2 \mathrm{f}(x)+2)]^2$ and $\mathrm{f}(0)=-1, \mathrm{f}^{\prime}(0)=1$ then $g^{\prime}(0)$ is

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^2-b^2$, where $\mathrm{a}>\mathrm{b}>0$, then $\frac{\mathrm{d} x}{\mathrm{~d} y}$ at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is

If

\(y=[(x+1)(2 x+1)(3 x+1) \ldots \ldots \ldots(n x+1)]^2\)

then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=0$ is

If $x=2 \cos \theta-\cos 2 \theta$ and $y=2 \sin \theta-\sin 2 \theta$, then $\frac{\mathrm{d}^2 y}{d x^2}$ is equal to

If $x=\sin \theta, y=\sin ^3 \theta$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ at $\theta=\frac{\pi}{2}$ is

The derivative of $\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)$ w.r.t. $\sin ^{-1}\left(3 x-4 x^3\right)$ is

If $y=\frac{x^{\frac{2}{3}}-x^{\frac{-1}{3}}}{x^{\frac{2}{3}}+x^{\frac{-1}{3}}}, x \neq 0$, then $(x+1)^2 y_1=$

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

If $y=\sin ^{-1}\left(\frac{\log x^2}{1+(\log x)^2}\right)$, then $\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{\mathrm{at ~}
x=1}=$

If $\mathrm{f}(x)=(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)$, then $f^{\prime}(1)=$

If $x^2+y^2=\mathrm{t}+\frac{1}{\mathrm{t}}, x^4+y^4=\mathrm{t}^2+\frac{1}{\mathrm{t}^2}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}=$

If $y=(\sin x)^{\tan x}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

Let $\mathrm{f}(x)=\frac{x}{\sqrt{\mathrm{a}^2+x^2}}-\frac{\mathrm{d}-x}{\sqrt{\mathrm{~b}^2+(\mathrm{d}-x)^2}}, x \in \mathbb{R}$ where $\mathrm{a}, \mathrm{b}, \mathrm{d}$ are non-zero real constants. Then

If $\mathrm{f}(x)=\log _{x^2}(\log x)$, then at $x=\mathrm{e}, \mathrm{f}^{\prime}(x)$ has the value

The curve $x^4-2 x y^2+y^2+3 x-3 y=0$ cuts the X -axis at $(0,0)$ at an angle of

If $y=a x^{n+1}+b x^{-n}$, then $x^2 \frac{d^2 y}{d x^2}=$

If $y$ is a function of $x$ and $\log (x+y)=2 x y$, then the value of $y^{\prime}(0)$ is

If \(y = {{\sin x} \over {1 + {{\cos x} \over {1 + {{\sin x} \over {1 + {{\cos x} \over {...}}}}}}}}\), then $\frac{dy}{dx}$ is given by

If $\frac{\mathrm{d}}{\mathrm{d} x} \mathrm{f}(x)=4 x^3-\frac{3}{x^4}$ such that $\mathrm{f}(2)=0$, then $\mathrm{f}(x)$ is equal to

If $y$ is a function of $x$ and $\log (x+y)=2 x y$, then the value of $y^{\prime}(0)$ is

If $x^2 y^2=\sin ^{-1} x+\cos ^{-1} x$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1$ and $y=2$ is

The derivative of \(\mathrm{f}(\sec x)\) with respect to \(g(\tan x)\) at \(x=\frac{\pi}{4}\), where \(f^{\prime}(\sqrt{2})=4\) and \(g^{\prime}(1)=2\), is

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

If \(x^2+y^2=\mathrm{t}+\frac{1}{\mathrm{t}}\) and \(x^4+y^4=\mathrm{t}^2+\frac{1}{\mathrm{t}^2}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to

The rate of change of \(\sqrt{x^2+16}\) with respect to \(\frac{x}{x-1}\) at \(x=5\) is

If \(\mathrm{g}\) is the inverse of \(\mathrm{f}\) and \(\mathrm{f}^{\prime}(x)=\frac{1}{1+x^3}\), then \(\mathrm{g}^{\prime}(x)\) is

If \(x^{\mathrm{k}}+y^{\mathrm{k}}=\mathrm{a}^{\mathrm{k}}(\mathrm{a}, \mathrm{k}>0)\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0\), then \(\mathrm{k}\) has the value

If \(y\) is a function of \(x\) and \(\log (x+y)=2 x y\), then the value of \(y^{\prime}(0)\) is

If \(y=[(x+1)(2 x+1)(3 x+1) \ldots \ldots(\mathrm{n} x+1)]^n\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

\(\text { If } y=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2, \text { then }\left(1-x^2\right) y_2-x y_1=\)

Let \(f: R \rightarrow R\) be a function such that \(\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+6, x \in \mathrm{R}\), then \(\mathrm{f}(2)\) equals