Methods of Differentiation

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

If $x^2+y^2=\mathrm{1,}$  then :

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

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

The function $f:R\to R$ satisfies $f\left(x^2\right)·f^{"}\left(x\right)=f^{\text{'}}\left(x\right)·f^{\text{'}}\left(x^2\right)$ for all real $x$ $.$ Given that $f\left(1\right)=1$ and $f^{\text{'}\text{'}\text{'}}\left(1\right)=8,$ compute the value of $f^{\text{'}}\left(1\right)+f^{"}\left(1\right).$

If $2x=y^{\frac{1}{5}}+\frac{1}{y^{\frac{1}{5}}}$ then $\left(x^2-1\right)\frac{{\text{d}}^{2}y}{\text{d}{x}^2}+x\frac{\text{d}y}{\text{d}x}=k\mathrm{y,}$then find the value of $k$

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 $y=e^x\left(\sin x+\cos x\right)$, then what would be the value of  $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y$ :

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:

If $y=3e^{2x}+2e^{3x}$, then what would be the value of given expression $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y$

If $y=a{e}^{bx}+b{e}^{ax}$ then find $y\text{'}\left(0\right)$

Find the set of points where $f\left(x\right)=x\left|x\right|$ is differentiable

Find $\frac{d}{dx}\left(\sin \left({\cot }^{-1}x\right)\right)$.