Differentiation of Composite & Inverse Functions (College Board AP® Calculus AB): Exam Questions

If $y={\left(x^4+\sin x\right)}^5$, then $y^{\text{'}}=$

Let $f\left(x\right)={\left(3x-1\right)}^3$ and let $g$ be the inverse function of $f$. Given that $f\left(0\right)=-1$, what is the value of $g^{\text{'}}\left(-1\right)$?

If $f\left(x\right)=\cos \left(e^{-2x}\right)$, then $f^{\text{'}}\left(x\right)=$

If $f\left(x\right)=\tan \left(2x\right)$, then $f^{\text{'}}\left(\frac{\pi }{2}\right)=$

A function $f$ is defined by $f\left(x\right)=\arccos \left(2x\right)$.

What is the second derivative of $f\left(x\right)$?

Let $f$ be the function defined by $f\left(x\right)=4x^2+2x$ for $x>-\frac{1}{4}$. If $g\left(x\right)=f^{-1}\left(x\right)$ and $g\left(20\right)=2$, what is the value of $g\text{'}\left(20\right)$?

If $f\left(x\right)=\sqrt{x^2-13}$ and $g\left(x\right)=4x+1$, then the derivative of $f\left(g\left(x\right)\right)$ at $x=1$ is

$\frac{d}{dx}\left(e^{e^x}\right)=$

If $f\left(x\right)=x^3+3x$, then $\frac{d}{dx}\left(f\left(\ln \left(x^2\right)\right)\right)=$

Let $f$ be a differentiable function such that $f\left(2\right)=-1$, $f\left(3\right)=2$, $f\text{'}\left(2\right)=-3$ and $f\text{'}\left(3\right)=-2.$

The function $g$ is differentiable and $g\left(x\right)=f^{-1}\left(x\right)$ for all $x.$

What is the value of $g\text{'}\left(2\right)?$

If $f\left(x\right)=\ln \left(2x-3+e^{-x}\right)$, then $f^{\text{'}\text{'}}\left(0\right)=$

An equation of the line tangent to the graph of $f\left(x\right)=x{\left(2-3x\right)}^4$ at the point $\left(1,1\right)$ is

If $f\left(x\right)=\left(x+3\right){\left(x^2-1\right)}^3$, then $f\text{'}\left(x\right)=$