Fundamental Properties of Differentiation (College Board AP® Calculus BC): Exam Questions

If $y=x\cos x$, then $\frac{dy}{dx}=$

If $f\left(x\right)=6-5x-\ln x$, then $f^{\text{'}}\left(1\right)=$

The function $f$ is defined by $f\left(x\right)=\frac{x}{x+3}$. What points $\left(x,y\right)$ on the graph of $f$ have the property that the line tangent to $f$ at $\left(x,y\right)$ has slope $\frac{1}{3}$?

Let $f$ and $g$ be differentiable functions with the following properties:

$f\left(x\right)>0$ for all $x$

$g\left(5\right)=3$

If $h\left(x\right)=f\left(x\right)g\left(x\right)$ and $h\text{'}\left(x\right)=f\text{'}\left(x\right)g\left(x\right)$, then $g\left(x\right)=$

If $f\left(x\right)=\tan \left(3x\right)$, then $f\text{'}\left(\frac{\pi }{18}\right)=$

An equation of the line tangent to the graph of $y=x+\sin x$ at the point $\left(0,0\right)$ is

If $y=\frac{x^2}{\ln x}$, then $\frac{dy}{dx}=$

If $f\left(x\right)=\frac{3x+2}{4x-7}$, then $f\text{'}\left(x\right)=$

If $y=x^3\cos 3x$, then $\frac{dy}{dx}=$

What is the instantaneous rate of change at $x=3$ of the function $f$ given by $f\left(x\right)=\frac{x^2-5}{x-4}$?

$\underset{h\to 0}{\lim }\frac{\tan 5\left(x+h\right)-\tan \left(5x\right)}{h}=$

If $f\left(x\right)=x\sqrt{2x}$, which of the following is not correct.

Let $f$ be a differentiable function with $f\left(3\right)=18$ and $f\text{'}\left(3\right)=9$, and let $g\left(x\right)=\ln xf\left(x\right)$. Which of the following is an equation of the line tangent to the graph of $g$ at the point where $x=3$?

For which value of $x$ do the graphs of $y={\left(\ln x\right)}^2$ and $y=\sqrt{x}+\frac{1}{e^x}$ have parallel tangent lines?