Derivatives of Composite and Implicit Functions

Differentiate $3^{e^x}$ with respect to $x$.

If $y={\sin }^{-1}\frac{2x}{1+x^2},$ find $\frac{dy}{dx}.$

If $y={\left({\cot }^{-1}{x}^3\right)}^2, \mathrm{find} \frac{dy}{dx}$

Find $\frac{dy}{dt}$, when $y={\sin }^{-1}\left(\frac{2\sqrt{t}-1}{t^2}\right)$

If $y=\sin \left({\log }_{e}x\right)$, prove that $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{x}$

Differentiate the following w.r.t. $x$.

${\sin }^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$ .

Differentiate the following w.r.t. $x$.

${\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right)$

Differentiate: $e^{\sqrt{x}}$ w.r.t. $\text{'}x\text{'}$.

Find $\frac{dy}{dx}$ if $y={\left(e^{secx}+x\right)}^4$. Show your work.

If $f\left(x\right)={\log }_e\left({\log }_{e}x\right),$ then $f^{\text{'}}\left(e\right)$ is equal to

Evaluate $\frac{d}{dx}\left({\sin }^{-1}2x\right)$

$\frac{d}{dx}\left(2\sin \frac{7x}{12}\right)=$

$y=\sqrt{1+x^2};$ prove that $y\text{'}=\frac{xy}{1+x^2}$

Find $\frac{dy}{dx}$ at $y={\tan }^{-1}\left(\frac{x}{a+\sqrt{a^2-x^2}}\right)$.

Find the derivative of $\sin t^2$.

Differentiate ${\tan }^{-1}\left(\frac{1}{\sqrt{1+x^2}+x}\right)$

Compute the derivative of the function $y=\arcsin (2x+1)$ i.e.,$y={\sin }^{-1}(2x+1)$.

If $x\sqrt{1+y}+y\sqrt{1+x}=0$, find $\frac{dy}{dx}$.

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

Differentiate the following w.r.t. $x$.

${\cos }^{-1}\left(\sin x\right)$