Rules of Differentiation

$\frac{d^{2}x}{d{y}^2}=$

If $\ln \left(x+y\right)=2xy$, then $y^{\text{'}}\left(0\right)$ is equal to

If $x^2+y^2=\mathrm{1,}$  then :

$If\text{f}\left(x\right)=\left|\begin{array}{ccc}\text{cos}\left(x+x^2\right)& sin\left(x+x^2\right)& -cos\left(x+x^2\right)\\ sin\left(x-x^2\right)& cos\left(x-x^2\right)& sin\left(x-x^2\right)\\ sin2x& 0& sin2x^2\end{array}\right|then,\; find { f}^{\prime }\left(x\right)$

$If\text{f}\left(x\right)=\left|\begin{array}{ccc}{\left(x-a\right)}^4& {\left(x-a\right)}^3& 1\\ {\left(x-b\right)}^4& {\left(x-b\right)}^3& 1\\ {\left(x-c\right)}^4& {\left(x-c\right)}^3& 1\end{array}\right|then{\text{f}}^{\prime }\left(x\right)=\lambda ·\left|\begin{array}{ccc}{\left(x-a\right)}^4& {\left(x-a\right)}^2& 1\\ {\left(x-b\right)}^4& {\left(x-b\right)}^2& 1\\ {\left(x-c\right)}^4& {\left(x-c\right)}^2& 1\end{array}\right|·\text{Find the value of}\lambda$

If the dependent variable y is changed to 'z' by the substitution y = tan z then the differential equation $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=1+\frac{2\left(1+y\right)}{1+y^2}{\left(\frac{\text{d}y}{\text{d}x}\right)}^2$ is changed to $\frac{{\text{d}}^{2}z}{\text{d}{x}^2}=cos{x}^{2}z+\text{k}{\left(\frac{\text{d}z}{\text{d}x}\right)}^2,$then find the value of k.

Let $P\left(x\right)$ be a polynomial of degree $4$ such that $P\left(1\right)=P\left(3\right)=P\left(5\right)=P^{\text{'}}\left(7\right)=0$. If the real number $x\ne 1,3,5$ is such that $P\left(x\right)=0$ can be expressed as $x=\frac{p}{q}$ where $\text{'}p\text{'} \text{and} \text{'}q\text{'}$ are relatively prime, then $\left(p+q\right)$ equals

A twice differentiable function $f\left(x\right)$ is defined for all real numbers and satisfies the following conditions, $f\left(0\right)=2; f^{\text{'}}\left(0\right)=-5 \text{and} f^{\text{'}\text{'}}\left(0\right)=3.$ The function $g\left(x\right)$ is defined by $$$g\left(x\right)=e^{ax}+f\left(x\right)\forall x\in R$, where $a$ is any constant. If $g^{\text{'}}\left(0\right)+g^{"}\left(0\right)=0.$ Then $a$ can be equal to 

if y = y(x) and it follows the relation $e^{xy}+ycosx=\mathrm{2,}$ then find (i) y' (0) and (ii) y'' (0)

Let $f\left(x\right)=x^2-4x-3, x>2$ and let $g\left(x\right)$ be the inverse of $f\left(x\right)$. Find the value of $g^{\text{'}}\left(2\right)$ where $f\left(x\right)=2$

Suppose $f\left(x\right)=\tan \left({\sin }^{-1}\left(2x\right)\right)$, then $f^{\text{'}}\left(\frac{1}{4}\right)$ is equal to

Find the derivative with respect to $x$ of the function :

$\left({\log }_{\cos x}\sin x\right){\left({\log }_{\sin x}\cos x\right)}^{-1}+arc\sin \frac{2x}{1+x^2}$ at $x=\frac{\pi }{4}$

Let$f\left(x\right)=x+\frac{1}{2x+\frac{1}{2x+\frac{1}{2x+\dots \dots \dots \infty }}}$ Then the value of$f\left(100\right)·f^{\text{'}}\left(100\right)$ is 

Differentiate $\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}$ w .r. t. $\sqrt{1-x^4}$

If $y={\log }_e{\left(x^{e^x·a^y}\right)}^{y^x}$, then $\frac{dy}{dx}$ is equal to

If $y=\frac{x^2}{2}+\frac{1}{2}x\sqrt{x^2+1}+\log \sqrt{x+\sqrt{x^2+1}}$, find the value of $x\frac{dy}{dx}+\log \frac{dy}{dx}$.

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