Differentiation · Mathematics · MHT CET

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

If $$\mathrm{f}^{\prime}(x)=\sin (\log x)$$ and $$y=\mathrm{f}\left(\frac{2 x+3}{3-2 x}\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ at $$x=1...

Let $$\mathrm{P}(x)$$ be a polynomial of degree 2, with $$\mathrm{P}(2)=-1, \mathrm{P}^{\prime}(2)=0, \mathrm{P}^{\prime \prime}(2)=2$$, then $$\mathr...

If $$y=\sqrt{(x-\sin x)+\sqrt{(x-\sin x)+\sqrt{(x-\sin x) \ldots.}}}$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}=$$

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}$...

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

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

The money invested in a company is compounded continuously. If ₹ 200 invested today becomes ₹ 400 in 6 years, then at the end of 33 years it will beco...

$$y=\frac{\sqrt[3]{1+3 x} \sqrt[4]{1+4 x} \sqrt[5]{1+5 x}}{\sqrt[7]{1+7 x} \sqrt[8]{1+8 x}} \text {. Then, } \frac{d y}{d x} \text { at } x=0$$ is...

Let $$f: R \rightarrow R$$ be a function such that $$f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R \text {, }...

If $$x=\log _e\left(\frac{\cos \frac{y}{2}-\sin \frac{y}{2}}{\cos \frac{y}{2}+\sin \frac{y}{2}}\right), \tan \frac{y}{2}=\sqrt{\frac{1-t}{1+t}}$$ Then...

Differentiation of $$\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$$ w.r.t. $$\cos ^{-1}\left(\sqrt{\frac{1+\sqrt{1+x^2}}{2 \sqrt{1+x^2}}}\right)$$ ...

If $$y=\tan ^{-1}\left(\frac{4 \sin 2 x}{\cos 2 x-6 \sin ^2 x}\right)$$, then $$\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)$$ at $$x=0$$ is

Let $$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$$ be a function such that $$\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \pri...

If $$\mathrm{f}(x)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right)$$, then $$\mathrm{f}^{\prime}(\mathrm{e})$$ is

For $$x>1$$, if $$(2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}$$, then $$\left(1+\log _e 2 x\right)^2 \frac{d y}{d x}$$ is equal to

If $$\tan y=\frac{x \sin \alpha}{1-x \cos \alpha}$$ and $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{m}}{x^2+2 \mathrm{n} x+1}$$, then $$\mathrm...

The derivative of $$\mathrm{f}(\tan x)$$ w.r.t. $$\mathrm{g}(\sec x)$$ at $$x=\frac{\pi}{4}$$ where $$\mathrm{f}^{\prime}(1)=2$$ and $$\mathrm{g}^{\pr...

If $$x=-1$$ and $$x=2$$ are extreme points of $$\mathrm{f}(x)=\alpha \log x+\beta x^2+x, \alpha$$ and $$\beta$$ are constants, then the value of $$\al...

$$\text { If } \log (x+y)=2 x y \text {, then } \frac{\mathrm{d} y}{\mathrm{~d} x} \text { at } x=0 \text { is }$$

$$y=(1+x)\left(1+x^2\right)\left(1+x^4\right) \ldots \ldots \ldots\left(1+x^{2 n}\right)$$, then the value of $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ a...

If $$\mathrm{f}(x)=3^x ; \mathrm{g}(x)=4^x$$, then $$\frac{\mathrm{f}^{\prime}(0)-\mathrm{g}^{\prime}(0)}{1+\mathrm{f}^{\prime}(0) \mathrm{g}^{\prime}...

$$\text { For all real } x \text {, the minimum value of } \frac{1-x+x^2}{1+x+x^2} \text { is }$$

The set of all points, where the derivative of the functions $$\mathrm{f}(x)=\frac{x}{1+|x|}$$ exists, is

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

If $$y=\log _{\sin x} \tan x$$, then $$\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{x=\frac{\pi}{4}}$$ has the value

Let $$\mathrm{f}(x)=\log (\sin x), 0 ...

Derivative of $$\tan ^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)$$ w.r.t. $$\cos ^{-1} x^2$$ is

Let $$f$$ be a differentiable function such that $$\mathrm{f}(1)=2$$ and $$\mathrm{f}^{\prime}(x)=\mathrm{f}(x)$$, for all $$x \in \mathrm{R}$$. If $$...

If $$y$$ is a function of $$x$$ and $$\log (x+y)=2 x y$$, then $$\frac{d y}{d x}$$ at $$x=0$$ is

If $$x=3 \tan \mathrm{t}$$ and $$y=3 \sec \mathrm{t}$$, then the value of $$\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$$ at $$\mathrm{t}=\frac{\pi}{4}$$ i...

If $$y=\tan ^{-1}\left(\frac{\log \left(\frac{\mathrm{e}}{x^2}\right)}{\log \left(e x^2\right)}\right)+\tan ^{-1}\left(\frac{4+2 \log x}{1-8 \log x}\r...

If $$y=\cos ^{-1}\left(\frac{\mathrm{a}^2}{\sqrt{x^4+\mathrm{a}^4}}\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ is

For $$x>1$$, if $$(2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}$$, then $$(1+\log 2 x)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}$$ is equal to

If $$\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x$$ and $$\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))$$, then $$\frac{\mathrm{h}^{\prime}(x)}{\...

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

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

The rate of change of $$\sqrt{x^2+16}$$ with respect to $$\frac{x}{x-1}$$ at $$x=5$$ 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 equ...

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...

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$...

If $$y=\log \sqrt{\frac{1+\sin x}{1-\sin x}}$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ at $$x=\frac{\pi}{3}$$ is

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

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

If $$y=1+x e^y$$, then $$\frac{d y}{d x}=$$

If $$x=e^t(\sin t-\cos t)$$ and $$y=e^t(\sin t+\cos t)$$, then $$\frac{d y}{d x}$$ at $$t=\frac{\pi}{3}$$ is

If $$\sin ^2 x+\cos ^2 y=1$$, then $$\frac{d y}{d x}=$$

$$ \text { If } u=\cos ^3 x, v=\sin ^3 x \text {, then }\left(\frac{d v}{d u}\right)_{x=\frac{\pi}{4}} \text { is equal to } $$

If $$y=\log _{10} x+\log _x 10+\log _x x+\log _{10} 10$$, then $$\frac{d y}{d x}=$$

If $$y=x \tan y$$, then $$\frac{d y}{d x}=$$

The derivative of the function $$\cot ^{-1}\left[(\cos 2 x)^{1 / 2}\right]$$ at $$x=\pi / 6$$ is

For all real $$x$$, the minimum value of the function $$f(x)=\frac{1-x+x^2}{1+x+x^2}$$ is

If $$f(x)=\operatorname{cosec}^{-1}\left[\frac{10}{6 \sin \left(2^x\right)-8 \cos \left(2^x\right)}\right]$$, then $$f^{\prime}(x)=$$

If $$y=\log \sqrt{\tan x}$$, then the value of $$\frac{d y}{d x}$$ at $$x=\frac{\pi}{4}$$ is

If $$\mathrm{x}=\mathrm{a}\left(\mathrm{t}-\frac{1}{\mathrm{t}}\right)$$ and $$\mathrm{y}=\mathrm{b}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)$$, th...

If $$y=\tan ^{-1} \sqrt{\frac{1+\cos x}{1-\cos x}}$$, then $$\frac{d y}{d x}=$$

If $$x^y \cdot y^x=16$$, then $\frac{d y}{d x}$ at $(2,2)$$ is

If $$y^2=a x^2+b x+c$$, where $$a, b, c$$ are constants, then $$y^3 \frac{d^2 y}{d x^2}$$ is equal to

$$x=\frac{1-t^2}{1+t^2}$$ and $$y=\frac{2 a t}{1+t^2}$$, then $$\frac{d y}{d x}=$$

If y = 2 sin x + 3 cos x and y + A$$\mathrm{\frac{d^2y}{dx^2}}$$ = B, then the values of A, B are respectively

If $$y = {\tan ^{ - 1}}\left\{ {{{a\cos x - b\sin x} \over {b\cos x + a\sin x}}} \right\}$$, then $${{dy} \over {dx}}$$

If $$e^{-y} \cdot y=x$$, then $$\frac{d y}{d x}$$ is

If $$y=\operatorname{cosec}^{-1}\left[\frac{\sqrt{x}+1}{\sqrt{x}-1}\right]+\cos ^{-1}\left[\frac{\sqrt{x}-1}{\sqrt{x}+1}\right]$$, then $$\frac{d y}{d...

The derivative of $$(\log x)^x$$ with respect to $$\log x$$ is

$$y=\sqrt{e^{\sqrt{x}}}$$, then $$\frac{d y}{d x}=$$

If $$y=\sin ^{-1}\left[\cos \sqrt{\frac{1+x}{2}}\right]+x^x$$, then $$\frac{d y}{d x}$$ at $$x=1$$ is

If $$x=a(t+\sin t), y=a(1-\cos t)$$, then $$\frac{d y}{d x}=$$

If $$y=\log \tan \left(\frac{x}{2}\right)+\sin ^{-1}(\cos x)$$, then $$\frac{d y}{d x}=$$

If $$h(x)=\sqrt{4 f(x)+3 g(x)}, f(1)=4, g(1)=3, f^{\prime}(1)=3, g^{\prime}(1)=4$$, then $$h^{\prime}(1)=$$

If $$x=a \cos \theta, y=b \sin \theta$$, then $$\left[\frac{d^2 y}{d x^2}\right]_{\theta=\frac{\pi}{4}}=$$

If $x=a \sin t-b \cos t, y=a \cos t+b \sin t$, then $y^3 \frac{d^2 y}{d x^2}+x^2+y^2=$

If $y=\sin ^{-1}\left[\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right]$, then $\frac{d y}{d x}=$

If $x^2 y^2=\sin ^{-1} \sqrt{x^2+y^2}+\cos ^{-1} \sqrt{x^2+y^2}$ then $\frac{d y}{d x}=$

If $$f(x)=\sin ^{-1}\left(\sqrt{\frac{1-x}{2}}\right)$$, then $$f^{\prime}(x)=$$

If $$\frac{x}{\sqrt{1+x}}+\frac{y}{\sqrt{1+y}}=0, x \neq y$$, then $$(1+x)^2 \frac{d y}{d x}=$$

If $$\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4$$, then $$\frac{d y}{d x}=$$

If $$f(x)=\log (\sec x+\tan x)$$, then $$f^{\prime}\left(\frac{\pi}{4}\right)=$$

If $y=\log \left[\frac{x+\sqrt{x^2+25}}{\sqrt{x^2+25}-x}\right]$ then $\frac{d y}{d x}=\ldots \ldots$

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

If $x^y=e^{x-y}$, then $\frac{d y}{d x}$ at $x=1$ is ...........

Derivative of $\log _{e^2}(\log x)$ with respect to $x$ is

If $x=\sqrt{a^{\sin ^{-1} t}}, y=\sqrt{a^{\cos ^{-1} t}}$, then $\frac{d y}{d x}=\ldots .$.