DERIVATIVE OF TRIGONOMETRIC FUNCTIONS — 32 Questions
32 multiple-choice questions on derivatives of trigonometric functions.
Try to solve each question independently before checking the answer key.
Instructions:
Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.
Click a selected answer again to deselect it.
Problem 1
Given the function $f(x)=\dfrac{2+\cos x}{\sin x}$. The tangent line at $x=\dfrac{\pi}{2}$ intersects the $y$-axis at $(0,b)$. The value of $b$ is...
A. $2$
B. $\dfrac{\pi}{2}$
C. $-2+\dfrac{\pi}{2}$
D. $2-\dfrac{\pi}{2}$
E. $2+\dfrac{\pi}{2}$
Problem 2
If $f(x)= -(\cos^{2}x-\sin^{2}x)$, then $f'(x)$ is...
A. $2(\cos x + \sin x)$
B. $2(\cos x - \sin x)$
C. $\sin x \cos x$
D. $2\sin x \cos x$
E. $4\sin x \cos x$
Problem 3
If $y=3x^{4}+\sin 2x + \cos 3x$, then $\dfrac{dy}{dx}=\cdots$
A. $12x^{3}+2\cos 2x +3\sin 3x$
B. $12x^{3}+\cos 2x - \sin 3x$
C. $12x^{3}-2\cos 2x +3\sin 3x$
D. $12x^{3}-2\cos 2x -3\sin 3x$
E. $12x^{3}+2\cos 2x -3\sin 3x$
Problem 4
If $y=2\sin 3x -3\cos 2x$, then $\dfrac{dy}{dx}=\cdots$
A. $2\cos 3x -3\sin 2x$
B. $6\cos 3x -3\sin 2x$
C. $2\cos 3x +3\sin 2x$
D. $6\cos 3x +6\sin 2x$
E. $-6\cos 3x - 6\sin 2x$
Problem 5
If $f(x)=\dfrac{\sin x+\cos x}{\sin x}$, $\sin x \neq 0$ and $f'(x)$ is the derivative of $f(x)$, then $f'\left(\dfrac{\pi}{2}\right)$ is...
A. $-2$
B. $-1$
C. $0$
D. $1$
E. $2$
Problem 6
If $f(x)=a\tan x + bx$, $f'\left(\dfrac{\pi}{4}\right)=3$ and $f'\left(\dfrac{\pi}{3}\right)=9$, then $a+b=\cdots$
A. $0$
B. $1$
C. $\dfrac{\pi}{2}$
D. $2$
E. $\pi$
Problem 7
The first derivative of $y=\cos^{4}x$ is...
A. $\dfrac{1}{4}\cos^{3}x$
B. $-\dfrac{1}{4}\cos^{3}x$
C. $\dfrac{1}{4}\sin^{3}x$
D. $-4\sin^{3}x \cos x$
E. $-4\cos^{3}x \sin x$
Problem 8
If $f(x)=a\tan x + bx$, $f'\left(\dfrac{\pi}{4}\right)=3$ and $f'\left(\dfrac{\pi}{3}\right)=9$, then $a+b=\cdots$
A. $0$
B. $2$
C. $\dfrac{24}{5}$
D. $6$
E. $\dfrac{39}{5}$
Problem 9
If $f(x)=\sin(\sin^{2}x)$, then $f'(x)=\ldots$
A. $2\sin x \cdot \cos(\sin^{2}x)$
B. $2\sin 2x \cdot \cos(\sin^{2}x)$
C. $\sin^{2}x \cdot \cos(\sin^{2}x)$
D. $\sin^{2}2x \cdot \cos(\sin^{2}x)$
E. $\sin 2x \cdot \cos(\sin^{2}x)$
Problem 10
Let $f(x)=2\tan(\sqrt{\sec x})$, then $f'(x)\cdots$
A. $\sec^{2}(\sqrt{\sec x}) \cdot \tan x$
B. $\sec^{2}(\sqrt{\sec x}) \cdot \sqrt{\sec x} \cdot \tan x$
C. $2\sec^{2}(\sqrt{\sec x}) \cdot \sqrt{\sec x} \cdot \tan x$
D. $\sec^{2}(\sqrt{\sec x}) \cdot \sec x \cdot \tan x$
E. $2\sec^{2}(\sqrt{\sec x}) \cdot \sec x \cdot \tan x$
Problem 11
The first derivative of $f(x)=\dfrac{1+\cos x}{\sin x}$ is $f'(x)=\cdots$
A. $\dfrac{1-\sin x}{\sin^{2}x}$
B. $\dfrac{\sin x-1}{\cos x-1}$
C. $\dfrac{2}{\cos x+1}$
D. $\dfrac{2}{\sin x-1}$
E. $\dfrac{1}{\cos x-1}$
Problem 12
If $f(x)=\sin ax + \cos bx$ satisfies $f'(0)=b$ and $f'\left(\frac{\pi}{2a}\right)=-1$, then $a+b=\cdots$
A. $-1$
B. $0$
C. $1$
D. $2$
E. $3$
Problem 13
If $f(x)=\sin x \cos 3x$, then $f'\left(\frac{1}{6}\pi\right)=\cdots$
A. $\frac{1}{2}$
B. $-\frac{1}{2}$
C. $-1\frac{1}{2}$
D. $-\frac{1}{2}+\sqrt{3}$
E. $-1\frac{1}{2}+\sqrt{3}$
Problem 14
The first derivative of $y=(\sin x + \cos x)^{2}$ is $y'=\cdots$
A. $0$
B. $4\sin^{2}x$
C. $4\sin^{2}x-2$
D. $4\cos^{2}x-2$
E. $4\cos^{2}x-4$
Problem 15
If $f(x)=\sqrt{1+\sin^{2}x}$, $0 \leq x \leq \pi$, then $f'(x) \cdot f(x)$ equals...
A. $(1+\sin^{2}x)\sin x \cos x$
B. $(1+\sin^{2}x)$
C. $\sin x \cos x$
D. $\sin x$
E. $\frac{1}{2}$
Problem 16
Given $f(x)=x\sin 3x$, then $f'\left(\frac{\pi}{4}\right)$ equals...
A. $\frac{\sqrt{2}}{2}\left(1+\frac{3\pi}{4}\right)$
B. $\frac{\sqrt{2}}{4}\left(1+\frac{3\pi}{4}\right)$
C. $\frac{\sqrt{2}}{2}\left(1-\frac{3\pi}{4}\right)$
D. $\frac{\sqrt{2}}{2}\left(\frac{3\pi}{4}-1\right)$
E. $-\frac{\sqrt{2}}{2}\left(1+\frac{3\pi}{4}\right)$
Problem 17
If $f(x)=\dfrac{\cos x -\sin x}{\cos x + \sin x}$, with $\cos x + \sin x \neq 0$, then $f'(x)=\cdots$
A. $1-(f(x))^{2}$
B. $-1+(f(x))^{2}$
C. $-(1+(f(x))^{2})$
D. $1+(f(x))^{2}$
E. $(f(x))^{2}$
Problem 18
If $f(x)=x\cos x$, then $f'\left(x+\frac{\pi}{2}\right)=\cdots$
A. $-\sin x - x\cos x + \frac{\pi}{2}\cos x$
B. $-\sin x - x\cos x - \frac{\pi}{2}\cos x$
C. $-\sin x + x\cos x - \frac{\pi}{2}\cos x$
D. $-\sin x + x\cos x + \frac{\pi}{2}\cos x$
E. $-\cos x + x\sin x + \frac{\pi}{2}\cos x$
Problem 19
Line $g$ is tangent to the curve $y=\sin x + \cos x$ at the point with abscissa $\frac{1}{3}\pi$. The gradient of the line perpendicular to $g$ is...
A. $1-\sqrt{3}$
B. $1+\sqrt{3}$
C. $1$
D. $\frac{\sqrt{3}-1}{2}$
E. $\frac{1-\sqrt{3}}{2}$
Problem 20
Given $f(x)=x^{\frac{1}{3}}\sin x$. The tangent line at $f$ that passes through the origin is...
A. $x=0$
B. $y=0$
C. $y=x$
D. $y=-x$
E. does not exist
Problem 21
If the tangent line to the curve $y=2x\cos^{3}x$ at $(\pi,-2\pi)$ is perpendicular to line $g$, then the equation of line $g$ is...
A. $y=2x-3\pi$
B. $y=2x+\pi$
C. $y=\frac{1}{2}x-\frac{5}{2}\pi$
D. $y=-\frac{1}{2}x+3\pi$
E. $y=\frac{1}{2}x+\pi$
Problem 22
Given $f(x)=\sin^{2}x$. If $f'(x)$ denotes the first derivative of $f(x)$, then $\lim\limits_{h \to \infty} h\left\{ f'\left(x+\frac{1}{h}\right)-f'(x)\right\}=\cdots$
A. $\sin 2x$
B. $-\cos 2x$
C. $2\cos 2x$
D. $2\sin x$
E. $-2\cos x$
Problem 23
If $f(x)=(\sin x + \cos x)(\cos 2x + \sin 2x)$ and $f'(x)=2\cos 3x + g(x)$, then $g(x)=\cdots$
A. $\cos 3x +\sin x$
B. $\cos 3x -\sin x$
C. $\cos x +\sin x$
D. $\cos x - \sin x$
E. $-\cos x + \sin x$
Problem 24
If $f(x)=2x+\sin 2x$ for $-\dfrac{\pi}{4} < x < \dfrac{\pi}{4}$, then $f'(x)=\cdots$
A. $4\sum\limits_{i=0}^{\infty}(\tan x)^{i}$
B. $4(1-\cos^{2}x)$
C. $4\sum\limits_{i=0}^{\infty}(-1)^{i}(\tan x)^{2i}$
D. $4\sum\limits_{i=0}^{\infty}(-\sin x)^{2i}$
E. $4\cos 2x$
Problem 25
The function $f(x)=-\sqrt{\cos^{2}x+\frac{x}{2}+\pi}$ for $-\pi < x < 2\pi$ is decreasing on the interval...
A. $0 < x < \dfrac{5\pi}{12}$
B. $0 < x < \dfrac{\pi}{12}$
C. $\dfrac{\pi}{6} < x < \dfrac{\pi}{3}$
D. $\dfrac{5\pi}{12} < x < \dfrac{7\pi}{12}$
E. $-\dfrac{7\pi}{12} < x < \dfrac{\pi}{12}$
Problem 26
The function $f(x)=\sqrt{\cos^{2}2x+x}$ for $x > 0$ is increasing on the interval...
A. $\dfrac{4\pi}{12} < x < \dfrac{13\pi}{12}$
B. $\dfrac{5\pi}{24} < x < \dfrac{13\pi}{24}$
C. $\dfrac{7\pi}{6} < x < \dfrac{11\pi}{6}$
D. $\dfrac{5\pi}{24} < x < \dfrac{11\pi}{24}$
E. $\dfrac{5\pi}{12} < x < \dfrac{11\pi}{12}$
Problem 27
The line perpendicular to the tangent line of $y=\tan x$ at $\left(\frac{\pi}{4},1\right)$ is...
A. $y=-\dfrac{x}{2}+\dfrac{\pi}{4}+1$
B. $y=-\dfrac{x}{2}+\dfrac{\pi}{8}-1$
C. $y=-\dfrac{x}{2}-\dfrac{\pi}{8}-1$
D. $y=-\dfrac{x}{2}-\dfrac{\pi}{4}-1$
E. $y=-\dfrac{x}{2}+\dfrac{\pi}{8}+1$
Problem 28
Given $f(x)=|\tan x|$, the rate of change of $f(x)$ at $x=k$, where $\dfrac{\pi}{2} < x < \pi$, is equal to...
A. $-\sin(k)$
B. $\cos(k)$
C. $-\sec^{2}(k)$
D. $\sec^{2}(k)$
E. $\cot(k)$
Problem 29
$y=\sin(\sin(\sin(\sin(\cdots(\sin(\sin x))\cdots))))$
Find $\dfrac{dy}{dx}$ at $x=0$.
A. $-\infty$
B. $-1$
C. $0$
D. $1$
E. $\infty$
Problem 30
The maximum value of $f(x)=2\cos 2x + 4\sin x$ for $0 < x < \pi$ is...
A. $2$
B. $3$
C. $4$
D. $-6$
E. $-12$
Problem 31
Given $f(x)=\dfrac{2+\cos x}{\sin x}$. The tangent line at $x=\dfrac{\pi}{2}$ intersects the $y$-axis at $(0,b)$, the value of $b$ is...
A. $2$
B. $\dfrac{\pi}{2}$
C. $-2+\dfrac{\pi}{2}$
D. $2-\dfrac{\pi}{2}$
E. $2+\dfrac{\pi}{2}$
Problem 32
The function $f(x)=\sqrt{2+\frac{x}{\sqrt{2}}-\sin x}$ for $-\pi \leq x \leq \pi$ is decreasing on the interval...
A. $0 \leq x \leq \frac{\pi}{2}$
B. $0 < x < \pi$
C. $-\frac{\pi}{3} \leq x \leq 0$
D. $-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}$
E. $-\frac{\pi}{4} < x < \frac{\pi}{4}$