Unit 2 test: Differentiation: Definition and Basic Rules

Find the average rate of change of $f(x) = x^3 - x$ on the interval $[-1, 2]$.

The table gives the height $H(t)$, in centimeters, of a plant $t$ days after it was planted.

Using the data, what is the best estimate of the rate at which the plant is growing at $t = 7$?

Evaluate $\displaystyle \lim_{h \to 0} \frac{\sqrt{4 + h} - 2}{h}$.

Use the definition of the derivative to find $f'(x)$ for $f(x) = 3x^2 - 5x$.

Let $f(x) = \begin{cases} x^2 + 2, & x \lt 1 \\ 4x - 1, & x \ge 1 \end{cases}$. Which statement is true?

Which of the following functions is not differentiable at $x = 0$?

Find $f'(x)$ for $f(x) = \dfrac{4}{x^2} + \sqrt{x}$.

Find all values of $x$ at which $f(x) = x^4 - 2x^2 + 7$ has a horizontal tangent line.

Find $f'(x)$ for $f(x) = 2\sin x + 3e^x - \ln x$.

Find an equation of the tangent line to $y = e^x + \cos x$ at $x = 0$.

Let $f(x) = x^2 \ln x$. Find $f'(e)$.

The functions $f$ and $g$ are differentiable, with $f(3) = 2$, $f'(3) = 4$, $g(3) = -1$ and $g'(3) = 5$. If $h(x) = \dfrac{f(x)}{g(x)}$, find $h'(3)$.

If $y = \dfrac{2x}{x + 3}$, then $\dfrac{dy}{dx} =$

Let $f(x) = \sec x + \tan x$. Find $f'(0)$.

$\displaystyle \lim_{h \to 0} \frac{\tan\left(\frac{\pi}{4} + h\right) - 1}{h}$ is