Fundamental Properties of Differentiation (College Board AP® Calculus AB): 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 $k$ be a differentiable function such that $k={\left(f\left(x\right)\right)}^2\times g\left(x\right)$.

Find the value of $k\text{'}\left(5\right)$.

Let function $f$ be a continuous function defined on the closed interval $0\le x\le 12$. The graph of $f$ consisting of three line segments is shown above. Let $G$ be the function defined by $G\left(x\right)={\int }_0^{x}f\left(t\right)\mathit{d}t$.

Let $P$ be the function defined by $P\left(x\right)=\frac{G\left(x\right)}{f\left(x\right)}$. Find $P^{\text{'}}\left(7\right)$.

The function $f$ is defined as $f\left(x\right)=\frac{10\sin x\cos x}{x^2-x+3}$.

Find the value of $f^{\text{'}}\left(0\right)$.

The function $f$ is defined as $f\left(x\right)=4^x\times e^{3x}$.

Find the slope of the tangent line to the graph of $y=f\left(x\right)$ at the point where $x=0$.

The differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f\text{'}$, $g$ and $g\text{'}$ for various values of $x$ are given in the table above.

Let $h\left(x\right)=\frac{f\left(x\right)}{2g\left(x\right)}$. Find $h\text{'}\left(-3\right)$.

Show the computations that lead to your answer.

The function $f$ is defined on the closed interval $\left[-6,6\right]$. The graph of $f$ consists of three line segments and a curve and is shown in the figure above. Let $g$ be the function defined by $g\left(x\right)={\int }_{-4}^{x}f\left(t\right)dt$.

The function $h$ is defined by $h\left(x\right)=\frac{2x^2}{g\left(x\right)}$. Find $h\text{'}\left(2\right)$.

Let $f$ be a differentiable function. The table above gives values of $f$ and its derivative $f\text{'}$ at selected values of $x$.

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

Let $h$ be the function defined by $h\left(x\right)=\frac{{\left(g\left(x\right)\right)}^2}{f\left(x\right)}$. Find $h\text{'}\left(-1\right)$.

The functions $f$ and $g$ are differentiable, where $f\left(x\right)={\sin }^{2}x$ and $g\left(x\right)=\sec x$.

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

Find the slope of the tangent line to the graph of $y=k\left(x\right)$ at the point where $x=\frac{\pi }{4}$.

The functions $f$ and $g$ are defined as $f\left(x\right)={\cos }^{2}x$ and $g\left(x\right)=\tan x$.

Let the function $h$ be defined by $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$. Find the value of $h^{\text{'}}\left(\frac{\pi }{3}\right)$.

The twice-differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f\text{'}$, $g$ and $g\text{'}$ for various values of $x$ are given in the table above.

The function $h$ is defined by $h\left(x\right)=\ln \left(5x\right)·f\left(x\right)+4g\left(x\right)$. Find $h\text{'}\left(\frac{1}{5}\right)$. Show the computations that lead to your answer.