Differentiation practice questions

Question 1

If $f(x) = x^3 - 3x^2 + 2$, at how many points is $f'(x) = 0$?

Accepted Answer

$f'(x) = 3x^2 - 6x = 3x(x-2)$. Roots: $x = 0$ and $x = 2$ — two critical points.

Answer

0

Answer

1

Answer

2

Answer

3

Question 2

What is $f'(x)$ if $f(x) = x^3$?

Accepted Answer

By the power rule, $	frac{d}{dx}\,x^3 = 3x^2$.

Answer

$x^2$

Answer

$3x^2$

Answer

$3x$

Answer

$x^3 \ln x$

Question 3

What is $f'(x)$ if $f(x) = \sin x + \cos x$?

Accepted Answer

Differentiating term by term: $(\sin x)' + (\cos x)' = \cos x - \sin x$.

Answer

$\sin x - \cos x$

Answer

$\cos x + \sin x$

Answer

$\cos x - \sin x$

Answer

$-\cos x - \sin x$

Question 4

If $f(x) = (x^2 + 1)^3$, find $f'(1)$.

Accepted Answer

$f'(x) = 3(x^2+1)^2 \cdot 2x = 6x(x^2+1)^2$. At $x=1$: $6 \cdot 1 \cdot 4 = 24$.

Answer

$12$

Answer

$18$

Answer

$24$

Answer

$48$

Question 5

Using the definition $f'(a) = \displaystyle\lim_{h	o 0} \dfrac{f(a+h) - f(a)}{h}$, the derivative of $f(x) = x^{2}$ at $x = 3$ is:

Accepted Answer

$\dfrac{f(3+h) - f(3)}{h} = \dfrac{(3+h)^{2} - 9}{h} = \dfrac{9 + 6h + h^{2} - 9}{h} = \dfrac{6h + h^{2}}{h} = 6 + h$.

$\displaystyle\lim_{h	o 0}(6 + h) = 6$.

Answer

$3$

Answer

$9$

Answer

$6$

Answer

$0$

Question 6

By Fermat's theorem on interior extrema, if $f(x) = x^{2} - 6x + 5$ attains its minimum on $\mathbb{R}$ at $x = c$, then $c$ equals:

Accepted Answer

$f'(x) = 2x - 6$.

Setting $f'(c) = 0$: $2c - 6 = 0 \implies c = 3$.

Since $f''(x) = 2 > 0$, this is indeed a minimum.

Answer

$-3$

Answer

$0$

Answer

$3$

Answer

$5$

Question 7

Apply the Mean Value Theorem to $f(x) = x^{2}$ on $[0, 4]$. The value $c \in (0,4)$ satisfying $f'(c) = \dfrac{f(4) - f(0)}{4 - 0}$ is:

Accepted Answer

Average rate: $\dfrac{f(4) - f(0)}{4 - 0} = \dfrac{16 - 0}{4} = 4$.

$f'(x) = 2x$, so $f'(c) = 4 \implies 2c = 4 \implies c = 2$.

$2 \in (0, 4)$ ✓.

Answer

$c = 4$

Answer

$c = 2$

Answer

$c = 1$

Answer

$c = 0$

Question 8

By Rolle's Theorem applied to $f(x) = x^{2} - 4$ on $[-2, 2]$, the guaranteed value $c \in (-2, 2)$ with $f'(c) = 0$ is:

Accepted Answer

$f$ is a polynomial (continuous on $[-2,2]$, differentiable on $(-2,2)$), and $f(-2) = 0 = f(2)$, so Rolle's hypotheses hold.

$f'(x) = 2x$, so $f'(c) = 0 \implies c = 0$.

Answer

$c = 1$

Answer

$c = 0$

Answer

$c = -1$

Answer

$c = 2$

Question 9

Compute $f'(x)$ for $f(x) = \ln(x^2 + 1)$.

Accepted Answer

$	frac{d}{dx}\,\ln u = 	frac{u'}{u}$. Here $u' = 2x$, so $f'(x) = \dfrac{2x}{x^2 + 1}$.

Answer

$\dfrac{1}{x^2 + 1}$

Answer

$\dfrac{2x}{x^2 + 1}$

Answer

$\dfrac{x}{x^2 + 1}$

Answer

$\dfrac{2}{x^2 + 1}$

Question 10

Compute $f'(x)$ for $f(x) = (x^2 + 1)^5$.

Accepted Answer

$f'(x) = 5(x^2 + 1)^4 \cdot 2x = 10x(x^2 + 1)^4$.

Answer

$5(x^2 + 1)^4$

Answer

$5x(x^2 + 1)^4$

Answer

$10x(x^2 + 1)^4$

Answer

$(2x)^5$

Differentiation

Problems & worked solutions

Problem 1 — Derivatives

Problem 2 — Differentiation

Problem 3 — Differentiation

Problem 4 — Chain Rule

Problem 5 — Definition of the Derivative

Problem 6 — Fermat's Theorem

Problem 7 — Mean Value Theorem

Problem 8 — Rolle's Theorem

Problem 9 — Differentiation

Problem 10 — Differentiation

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