Differentiation

If $y=\left(x+\sqrt{1+x^2}\right)^n$, then $\left(1+x^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is

Let $f\left(x\right)=\left(\sin\,\left(\tan\,^{-1}x\right)+\sin\,\left(\cot\,^{-1}x\right)\right)^{2}-1$,$\left|x\right|>1$. If $\frac{dy}{dx}=\frac{1}{2}\frac{d}{dx}\left(\sin\,^{-1}\left(f\left(x\right)\right)\right)$…

If $y=y(x)$ is an implicit function of $x$ such that $\log\,_{e}(x+y)=4xy$, then $\frac{d^{2}y}{dx^{2}}$ at $x=0$is equal to

Let $f(x)=x^{3}+x^{2}f^{'}(1)+xf^{"}(2)+f^{\operatorname{'''}\,}(3),x\in\,R$. Then $f^{'}(10)$ is equal to

If $\cos\,^{-1}\left(\frac{y}{2}\right)=\log\,_{e}\left(\frac{x}{5}\right)^{5},\left|y\right|<2$, then

For the curve $C:\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0$, the value of $3y^{'}-y^{3}y^{''}$, at the point $\left(\alpha\,,\alpha\,\right),\alpha\,>0$, on $C$, is equal to ________.

If $f\left(x\right)=\sin\,\left(\cos\,^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)\right)$ and its first derivative with respect to $x$ is $-\frac{b}{a}\log\,_{e}2$when $x=1,$ where $a$ and $b$ are integers, then the…

Let $f(x)=\cos\,\left(2\tan\,^{-1}\sin\,\left(\cot\,^{-1}\sqrt{\frac{1-x}{x}}\right)\right),0<x<1$. Then:

Let, $f:R\rightarrow\,R$ be a function such that $f\left(x\right)=x^{3}+x^{2}f'\left(1\right)+xf\operatorname{''}\,\left(2\right)+f\operatorname{'''}\,\left(3\right),\forall\,x\in\,R.$Then $f\left(2\right)$ equals

If $y=\left(x+\sqrt{1+x^2}\right)^n$, then $\left(1+x^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbf{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ is equal to :

Let $f$ be a real polynomial of degree $n$ such that $f(x) = f'(x) f''(x)$, for all $x \in \mathbb{R}$. If $f(0) = 0$, then $36\left(f'(2) + f''(2) + \int_0^2 f(x)\,dx\right)$ is equal to: