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

The functions $f$ and $g$ are differentiable. The table shown gives the values of the functions and their first derivatives at selected values of $x$.

Let $h$ be a differentiable function such that $h\left(x\right)=f\left(g\left(x\right)\right)$.

Find the value of $h\text{'}\left(3\right)$.

The function $f$ is defined by $f\left(x\right)=\sqrt{64-x^2}$ for $-8\le x\le 8$.

Find $f^{\text{'}}\left(x\right)$.

Write an equation for the line tangent to the graph of $f$ at $x=2\sqrt{7}$.

Let $f$ be the function defined by $f\left(x\right)=e^{\cos x}-\sin 2x$.

Find the slope of the line tangent to the graph of $f$ at $x=2\pi$.

Let $g$ be the function whose graph, consisting of three line segments, is shown in the figure below.

Let $h$ be the function defined by $h\left(x\right)=g\left(f\left(x\right)\right).$ Find $h^{\text{'}}\left(2\pi \right).$

Let $f$ be the function defined by $f\left(x\right)=\sin \left(\sqrt{x^2+1}\right)$.

Find the equation of the line tangent to the graph of $f$ at $x=0.$

The function $f$ is defined by $f\left(x\right)={\left(2x-5\right)}^3.$

Show that the inverse of $f\left(x\right)$ exists and find the derivative of the inverse of $f\left(x\right)$ at the point where $x=-27.$

Show all necessary steps involved in solving this problem.

The functions $f$ and $g$ are twice differentiable. The table gives values of the functions and their first derivatives at selected values of $x$.

Let $h$ be the function defined by $h\left(x\right)=f\left(g\right(x\left)\right)$. Find $h\text{'}\left(7\right)$. Show the work that leads to your answer.

Let $f$ be the function defined by $f\left(x\right)=\frac{3}{2x^2-7x+5}$.

Find the slope of the line tangent to the graph of $f$ at $x=3$.

Consider the family of functions $f\left(x\right)=\frac{1}{x^2-2x+k}$, where $k$ is a constant.

Find the value of $k$, for $k>0$, such that the slope of the line tangent to the graph of $f$ at $x=0$ equals $6$.

The function $f$ is defined as $f\left(x\right)=\frac{x}{2\pi }\cos \left(\frac{x^2}{\pi }\right)$.

$f^{\text{'}}\left(\frac{\pi }{2}\right)$ can be written as $\frac{\sqrt{a}\left(b-\pi \right)}{c\pi }$ where $a,b$ and $c$ are integers. Find the values of $a,b$ and $c$.

The function $f$ is defined as $f\left(x\right)=\arcsin \left(2x\right)$ for $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$.

Find the exact value of $f^{\text{'}}\left(\frac{1}{4}\right)$.

Let $f$ be the function defined by $f\left(x\right)=k\sqrt{3x+1}-\ln \left(2x\right)$ for $x>0,$ where $k$ is a positive constant.

Find $f^{\text{'}}\left(x\right)$ and $f^{\text{'}\text{'}}\left(x\right).$

Let $f$ be the function defined by $f\left(x\right)={\sin }^2\left(e^{x^2}\right).$

Differentiate $f\left(x\right)$ with respect to $x,$ showing all working clearly.

Let $g\left(x\right)=\ln f\left(x\right),$ where $f\left(x\right)={\sin }^2\left(e^{x^2}\right).$

Show that $g^{\text{'}}\left(x\right)\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & 4\end{array}\begin{array}{rcl}& & x\end{array}{e}^{x^2}\begin{array}{rcl}& & \cot \end{array}\begin{array}{rcl}& & \left(e^{x^2}\right)\end{array}.$

A roller coaster travels along a track, and the height of the coaster above the ground at time $t$ seconds is modeled by the function $h\left(t\right)=\tan \left(\ln \left(t^2+1\right)\right),$ where $t$ represents the time in seconds since the coaster enters a section of the track just before it hits a water feature. The model is valid for $-1\le t\le 1,$representing a brief window of time just before and after the coaster splashes into the water.

Determine the value of $h^{\text{'}}\left(t\right),$ and explain what it tells us about the roller coaster's velocity at the moment it hits the water.

Determine the value of $h^{\text{'}\text{'}}\left(0\right)$ and use this to describe the change in the behavior of the roller coaster's height at the moment it hits the water.

Let $f$ be the function defined by $f\left(x\right)=\text{cos}\left(2x\right)+e^{\text{sin}x}$. Let $g$ be a differentiable function. The table below gives values of $g$ and its derivative $g\text{'}$ at selected values of $x$.

Let $h$ be the function whose graph, consisting of five line segments, is shown below.

Find the slope of the line tangent to the graph of $f$ at $x=\pi$.

Let $k$ be the function defined by $k\left(x\right)=h\left(f\right(x\left)\right)$. Find $k\text{'}\left(\pi \right)$.

Let $m$ be the function defined by $m\left(x\right)=g(-2x) h\left(x\right)$. Find $m\text{'}\left(2\right)$.

The functions $f$ and $g$ have continuous second derivatives. The table below gives values of the functions and their derivatives at selected values of $x$.

Let $k\left(x\right)=f\left(g\left(x\right)\right)$. Write an equation for the line tangent to the graph of $k$ at $x=3$.