Differentiation: Definition & Basic Rules

The average rate of change of $f(x)=x^{2}+2x$ on $[1,3]$ is

• (A) $4$
• (B) $6$
• (C) $8$
• (D) $10$

$\displaystyle\lim_{h\to 0}\dfrac{(3+h)^{2}-9}{h}$ equals

• (A) $0$
• (B) $3$
• (C) $6$
• (D) $9$

If $f$ is differentiable at $x=a$, which of the following must be true?

• (A) $f$ is continuous at $x=a$.
• (B) $f$ has a local extremum at $x=a$.
• (C) $\displaystyle\lim_{x\to a}f(x)=0$.
• (D) $f''(a)$ exists.

Which of the following is NOT differentiable at $x=0$?

• (A) $f(x)=x^{2}$
• (B) $f(x)=|x|$
• (C) $f(x)=\sin x$
• (D) $f(x)=x^{3}$

If $f(x)=5x^{4}-3x^{2}+7$, then $f'(x)=$

• (A) $20x^{3}-6x$
• (B) $20x^{3}-6x+7$
• (C) $5x^{3}-3x$
• (D) $20x^{3}-3x^{2}$

$\dfrac{d}{dx}\bigl[\sin x - \cos x\bigr]=$

• (A) $\cos x - \sin x$
• (B) $\cos x + \sin x$
• (C) $-\cos x - \sin x$
• (D) $-\sin x - \cos x$

$\dfrac{d}{dx}\bigl[\,e^{x}\ln x\,\bigr]=$

• (A) $\dfrac{e^{x}}{x}$
• (B) $e^{x}\ln x+\dfrac{e^{x}}{x}$
• (C) $e^{x}\ln x$
• (D) $e^{x}+\ln x$

If $h(x)=x^{2}\cos x$, then $h'(x)=$

• (A) $2x\cos x - x^{2}\sin x$
• (B) $2x\cos x + x^{2}\sin x$
• (C) $-2x\sin x$
• (D) $2x - \sin x$

If $g(x)=\dfrac{x}{x^{2}+1}$, then $g'(x)=$

• (A) $\dfrac{1}{2x}$
• (B) $\dfrac{1-x^{2}}{(x^{2}+1)^{2}}$
• (C) $\dfrac{x^{2}+1-2x^{2}}{x^{2}+1}$
• (D) $\dfrac{x^{2}-1}{(x^{2}+1)^{2}}$

The tangent line to $f(x)=x^{3}-2x$ at $x=1$ has equation

• (A) $y=x-2$
• (B) $y=x-1$
• (C) $y=x-3$
• (D) $y=-x+1$

Let $f(x)=\dfrac{x\sin x}{e^{x}}$. Which differentiation rules are required?

• (A) Product only
• (B) Quotient only
• (C) Product and quotient
• (D) None (power rule suffices)

Selected values of $f,g,f',g'$ at $x=2$ are given.

If $h(x)=\dfrac{f(x)}{g(x)}$, then $h'(2)=$

• (A) $\dfrac{17}{16}$
• (B) $\dfrac{23}{16}$
• (C) $\dfrac{13}{4}$
• (D) $\dfrac{19}{16}$

The normal line to $y=\sqrt{x}$ at $x=4$ has slope

• (A) $4$
• (B) $-4$
• (C) $\dfrac{1}{4}$
• (D) $-\dfrac{1}{4}$

If $f(x)=\sin(2x)$, then $f^{(4)}(x)=$

• (A) $16\sin(2x)$
• (B) $-16\sin(2x)$
• (C) $16\cos(2x)$
• (D) $-8\sin(2x)$

A particle's position is $s(t)=t^{3}-6t^{2}+9t$. The acceleration at $t=3$ is

• (A) $0$
• (B) $3$
• (C) $6$
• (D) $12$

The graph of $f$ consists of two line segments meeting at a sharp corner at $x=3$. Which statement about $f'$ is true?

• (A) $f'(3)$ exists and equals zero.
• (B) $f'$ has a jump at $x=3$; $f'(3)$ does not exist.
• (C) $f$ is not continuous at $x=3$.
• (D) $f'$ is constant on $(1,5)$.

$\dfrac{d}{dx}\bigl[\,3^{x}\,\bigr]=$

• (A) $3^{x}$
• (B) $x\cdot 3^{x-1}$
• (C) $3^{x}\ln 3$
• (D) $\dfrac{3^{x}}{\ln 3}$

Let $f(x)=x\,e^{-x}$. The value of $x>0$ at which the tangent line to $y=f(x)$ is horizontal is closest to

• (A) $0.5$
• (B) $1.0$
• (C) $1.5$
• (D) $2.0$

$\displaystyle\lim_{h\to 0}\dfrac{\sin\!\left(\tfrac{\pi}{3}+h\right)-\sin\!\left(\tfrac{\pi}{3}\right)}{h}=$

• (A) $0$
• (B) $\dfrac{1}{2}$
• (C) $\dfrac{\sqrt{3}}{2}$
• (D) $1$

$\displaystyle\lim_{h\to 0}\dfrac{e^{2+h}-e^{2}}{h}=$

• (A) $0$
• (B) $1$
• (C) $e$
• (D) $e^{2}$

Consider $f(x)=x^{2/3}$. Which statement about $f$ at $x=0$ is true?

• (A) $f$ is differentiable at $x=0$ with $f'(0)=0$.
• (B) $f$ has a vertical tangent at $x=0$ — the one-sided limits of $f'$ are both $+\infty$.
• (C) $f$ has a cusp at $x=0$ — $f$ is continuous there, but the one-sided limits of $f'$ at $0$ are infinite with opposite signs.
• (D) $f$ is not continuous at $x=0$.