2. Derivatives

Derivative Definition — Quiz

Test your understanding of derivative definition with 5 practice questions.

Practice Questions

Question 1

Given a function $f(x)$ that is differentiable at $x=a$, which of the following statements about the existence of the limit of the difference quotient is true?

Question 2

Consider a function $f(x)$ and a point $x=a$. If the instantaneous rate of change of $f(x)$ at $x=a$ is $-3$, what can be concluded about the behavior of the function at that point?

Question 3

The limit definition of the derivative is given by $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. If $f(x) = \sqrt{x}$, which of the following is the correct first step in evaluating its derivative using this definition?

Question 4

A particle's position is described by the function $s(t) = t^3 - 6t^2 + 9t$, where $t$ is in seconds and $s(t)$ is in meters. What is the instantaneous velocity of the particle at $t=2$ seconds?

Question 5

If a function $g(x)$ is differentiable at a point $x=c$, which of the following must be true about the tangent line to the graph of $g(x)$ at $x=c$?