The definition of the derivative

Which limit represents $g'(9)$ for $g(x) = \sqrt{x}$?

A particle moves along a line with position $s(t) = t^{2} + 4t$ metres at time $t$ seconds. Find its average velocity over the interval from $t = 1$ to $t = 4$.

Let $T(d)$ be the temperature of a lake, in degrees Celsius, at a depth of $d$ metres below the surface on a summer afternoon. What are the units of $T'(d)$, and what sign would you expect it to have?

Use the limit definition of the derivative to find $f'(2)$ for $f(x) = x^{2} - 4x$.

Use the definition of the derivative to find $f'(1)$ for the function below.

Use the definition of the derivative to find $f'(4)$ for the function below.

Let $f(x) = |x - 2|$. Which statement about $f$ at $x = 2$ is correct?

Find the values of the constants $a$ and $b$ for which the function below is differentiable everywhere, and enter $a + b$.

A ball is thrown straight up from a balcony. Its height in metres after $t$ seconds is $h(t) = 1.2 + 14.7t - 4.9t^{2}$. Find the maximum height the ball reaches.

Let $f(x) = x^{2}\sin\!\left(\dfrac{1}{x}\right)$ for $x \neq 0$ and $f(0) = 0$. Using the definition of the derivative, show that $f'(0)$ exists and find its value.