Methods 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 :

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

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

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 

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

If  $x=a\left(\cos t+t\sin t\right)$ and  $y=a\left(\sin t-t\cos t\right), 0<t<\frac{\pi }{2}.$ The value of   $\frac{d^{2}y}{d{x}^2}$ would be:

Derivative of ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$ with respect to ${\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ in $\left(1,\infty \right)$, is

The derivative of $a^{\sec x}$ w.r.t. $a^{\tan x} \left(\forall a>0\right)$ is

If $x={\cos }^3\theta$ and $y={\sin }^3\theta$, then $1+{\left(\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)}^2$ is equal to

The derivative of ${\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}x-\mathrm{c}\mathrm{o}\mathrm{s}x}{\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}x}\right)$ with respect to $\frac{x}{2},$ where $x\in \left(0,\frac{\pi }{2}\right)$, is