Question 1

Given the function $f(x) = x^5 - 3x^4 + x$, what is the third derivative $f^{(3)}(x)$?

Accepted Answer

$60x^2 - 72x$

Question 2

Which of the following best describes the physical meaning of the second derivative of a position function $s(t)$?

Accepted Answer

It represents the acceleration of the object.

Question 3

If $y = \frac{1}{x}$, what is $\frac{d^2y}{dx^2}$?

Accepted Answer

$2x^{-3}$ or $\frac{2}{x^3}$

Question 4

For $y = (x^2 + 10x)^{20}$, what is $y'$ using the chain rule?

Accepted Answer

$20(x^2 + 10x)^{19}(2x + 10)$

Question 5

Given $s(t) = 10t^2$, where $s$ is in miles and $t$ in hours, what is the velocity $v(4)$ when $t = 4$ hours?

Accepted Answer

$80$ mi/hr

Question 6

What is the fourth derivative $\frac{d^4}{dx^4}\left[\frac{1}{x}\right]$?

Accepted Answer

$\frac{24}{x^5}$

Question 7

A pebble is dropped from a hot-air balloon. If $s(t) = 16t^2$, what is its acceleration after $3.5$ seconds?

Accepted Answer

$32$ 	ext{ feet/second}^2$

Question 8

Which notation correctly represents the $n$th derivative of a function $f(x)$?

Accepted Answer

$f^{(n)}(x)$ or \frac{d^n}{dx^n}f(x)$

Question 9

If $y = -6x^4 + 3x^2$, what is $y''$?

Accepted Answer

$-72x^2 + 6$

Question 10

Which of the following is NOT a correct interpretation of the second derivative $f''(x)$?

Accepted Answer

It represents the total area under the curve $f(x)$.

Question 11

Given $f(x) = x^5 - 3x^4 + x$, what is $f^{(6)}(x)$?

Accepted Answer

$0$

Question 12

If $s(t)$ is the distance from a starting point at time $t$, which formula gives the velocity $v(t)$?

Accepted Answer

$v(t) = s'(t)$

Question 13

For $y = (3x^2 + 1)^2$, what is $y''$?

Accepted Answer

$78x^2 + 12$

Question 14

Which of the following is the correct Leibniz notation for the second derivative of $y$ with respect to $x$?

Accepted Answer

$\frac{d^2y}{dx^2}$

Question 15

If $s(t) = 16t^2$, what is the velocity $v(3.5)$ after $3.5$ seconds?

Accepted Answer

$112$ 	ext{ feet/second}$

Question 16

If $f(x) = x^5 - 3x^4 + x$, what is $f^{(4)}(x)$?

Accepted Answer

$120x - 72$

Question 17

Which of the following statements is true about higher-order derivatives?

Accepted Answer

The $n$th derivative of a polynomial of degree $m$ is zero for all $n > m$.

Higher Order Derivatives: Concepts, Notation, and Applications

Study Guide - Practice Questions