AP Calculus AB Unit 3: Composite, Implicit and Inverse Functions

The chain rule, implicit differentiation, the derivative of an inverse function from a formula or a table, the derivatives of the inverse trig functions, choosing a method, and higher-order derivatives (including second derivatives of implicit relations).

CED topics in this unit: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6.

Key formulas and facts

• Chain rule: $\frac{d}{dx}f(g(x))=f'(g(x))\,g'(x)$
• Inverse: if $g=f^{-1}$ and $f(a)=b$, then $g'(b)=\frac{1}{f'(a)}$
• $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$, $\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}$, $\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$
• $\frac{d}{dx}a^x=a^x\ln a$ and $\frac{d}{dx}\log_ax=\frac{1}{x\ln a}$
• Implicit: differentiate both sides with respect to $x$; every term in $y$ picks up a factor $\frac{dy}{dx}$
• $\frac{d^2y}{dx^2}$ is the derivative of $\frac{dy}{dx}$; substitute $\frac{dy}{dx}$ back in

Worked examples

• $\frac{d}{dx}\sin^3(2x)=3\sin^2(2x)\cos(2x)\cdot2=6\sin^2(2x)\cos(2x)$
• Implicit: $x^2+xy+y^3=11$ gives $2x+y+x\frac{dy}{dx}+3y^2\frac{dy}{dx}=0$, so $\frac{dy}{dx}=-\frac{2x+y}{x+3y^2}$; at $(1,2)$ the slope is $-\frac{4}{13}$
• Inverse from a table: $f(3)=7$ and $f'(3)=\frac12$, so $(f^{-1})'(7)=2$
• $\frac{d}{dx}\arctan(x^2)=\frac{2x}{1+x^4}$
• Second derivative: $y=e^{-x^2}$ gives $y'=-2xe^{-x^2}$ and $y''=(4x^2-2)e^{-x^2}$

Common traps

• Dropping the inner derivative in the chain rule.
• Evaluating the inverse derivative at the wrong point: $(f^{-1})'(b)$ needs $f'(a)$ where $f(a)=b$, not $f'(b)$.
• Forgetting $\frac{dy}{dx}$ on $y$ terms, or the product rule on $xy$, in implicit differentiation.
• Reading $\sin^{-1}x$ as $\frac{1}{\sin x}$: it means $\arcsin x$.
• Leaving $\frac{dy}{dx}$ inside a second derivative instead of substituting for it.

Practice questions

40 multiple-choice and 4 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

If $y=\sqrt{5x^2+4}$, what is $\dfrac{dy}{dx}$ at $x=1$?

1. $\frac{1}{6}$
2. $\frac{5}{3}$
3. $\frac{10}{3}$
4. $\frac{5}{6}$

Question 2 · Multiple choice

The table gives values of $f$, $f'$, $g$ and $g'$. If $h(x)=f(g(x))$, what is $h'(1)$?

1. $-18$
2. $12$
3. $-12$
4. $-21$

Question 3 · Multiple choice

The point $(0,3)$ lies on the curve $y^2+3xy=x^2+9$. What is $\dfrac{dy}{dx}$ at $(0,3)$?

1. $- \frac{3}{2}$
2. $\frac{3}{2}$
3. $- \frac{2}{3}$
4. $0$

Question 4 · Multiple choice

Let $f(x)=x^3+3x+2$ and let $g$ be the inverse function of $f$. What is $g'(6)$?

1. $\frac{1}{111}$
2. $111$
3. $6$
4. $\frac{1}{6}$

Question 5 · Multiple choice

What is the derivative of $\arcsin(3x)$ at $x=\dfrac16$?

1. $2 \sqrt{3}$
2. $\frac{2 \sqrt{3}}{3}$
3. $6$
4. $- 2 \sqrt{3}$

Question 6 · Multiple choice

If $y=\ln(x^2+1)$, what is $\dfrac{d^2y}{dx^2}$ at $x=2$?

1. $\frac{4}{5}$
2. $\frac{6}{25}$
3. $- \frac{6}{25}$
4. $- \frac{2}{25}$

Question 7 · Multiple choice

What is the derivative of $3^{x^2}$ at $x=1$?

1. $3 \log{\left(3 \right)}$
2. $6$
3. $6 \log{\left(3 \right)}$
4. $2$

Question 8 · Multiple choice

The point $(2,2)$ lies on the curve $x^2+4y^2=20$. What is $\dfrac{d^2y}{dx^2}$ at $(2,2)$?

1. $\frac{5}{32}$
2. $- \frac{1}{4}$
3. $- \frac{1}{16}$
4. $- \frac{5}{32}$

Question 9 · Free response (9 points)

Consider the curve given by $x^2-4xy+5y^2=1$.

1. (a) Show that $\dfrac{dy}{dx}=\dfrac{2y-x}{5y-2x}$. [2 points]
2. (b) Find the $y$-coordinates of the two points on the curve where $x=1$, and find the slope of the curve at each. [3 points]
3. (c) Find the coordinates of each point on the curve where the tangent line is vertical. [2 points]
4. (d) The point $(2,1)$ is on the curve. Find $\dfrac{d^2y}{dx^2}$ at $(2,1)$. What does it tell you about the curve at that point? [2 points]

Question 10 · Free response (9 points)

The function $f$ is differentiable, and $f'(x)>0$ for all $x$. The table gives selected values of $f$ and $f'$. Let $g$ be the inverse function of $f$.

1. (a) Explain why $f$ has an inverse function, and find $g'(3)$. [2 points]
2. (b) Let $h(x)=\big(f(x)\big)^3$. Find $h'(2)$. [2 points]
3. (c) Let $k(x)=\arctan\big(f(x)\big)$. Find $k'(3)$. [2 points]
4. (d) Let $p(x)=f\big(f(x)\big)$. Find $p'(2)$. [2 points]
5. (e) Write an equation of the line tangent to the graph of $g$ at $x=3$. [1 point]

Question 11 · Multiple choice

If $y=(3x^2-1)^4$, what is $\dfrac{dy}{dx}$ at $x=1$?

1. $32$
2. $96$
3. $192$
4. $384$

Question 12 · Multiple choice

If $f(x)=e^{x^2-3x}$, what is $f'(3)$?

1. $1$
2. $0$
3. $3 e$
4. $3$

Question 13 · Multiple choice

The table gives values of differentiable functions $f$ and $g$ and their derivatives. If $h(x)=g\big(f(x)\big)$, what is $h'(1)$?

1. $-6$
2. $4$
3. $12$
4. $15$

Question 14 · Multiple choice

If $y=\ln(\cos x)$, what is $\dfrac{dy}{dx}$ at $x=\frac{\pi}{4}$?

1. $1$
2. $- \sqrt{2}$
3. $-1$
4. $\frac{\sqrt{2}}{2}$

Question 15 · Multiple choice

The point $(1,2)$ lies on the curve $2x^2+xy+y^2=8$. What is $\dfrac{dy}{dx}$ there?

1. $- \frac{5}{6}$
2. $\frac{6}{5}$
3. $- \frac{2}{5}$
4. $- \frac{6}{5}$

Question 16 · Multiple choice

For the curve $x^3+y^3=9$, what is the slope of the tangent line at the point $(1,2)$?

1. $- \frac{1}{4}$
2. $-4$
3. $\frac{1}{4}$
4. $- \frac{1}{2}$

Question 17 · Multiple choice

If $xy^2=4$, what is $\dfrac{dy}{dx}$ at the point $(1,2)$?

1. $1$
2. $-1$
3. $-2$
4. $- \frac{1}{2}$

Question 18 · Multiple choice

At which points on the curve $x^2-xy+y^2=3$ is the tangent line horizontal?

1. $(1,2)$ and $(-1,-2)$
2. $(2,1)$ and $(-2,-1)$
3. $(\sqrt3,0)$ and $(-\sqrt3,0)$
4. $(1,1)$ and $(-1,-1)$

Question 19 · Multiple choice

The function $g$ is the inverse of $f(x)=x^3+3x-2$. What is $g'(2)$?

1. $6$
2. $\frac{1}{15}$
3. $\frac{1}{2}$
4. $\frac{1}{6}$

Question 20 · Multiple choice

The function $f$ is differentiable and invertible; the table gives selected values of $f$ and $f'$. If $g$ is the inverse of $f$, what is $g'(5)$?

1. $\frac{1}{4}$
2. $4$
3. $\frac{1}{3}$
4. $\frac{1}{2}$

Question 21 · Multiple choice

Let $f(x)=e^{2x}+x$, and let $g$ be the inverse of $f$. What is $g'(1)$?

1. $3$
2. $\frac{1}{2}$
3. $\frac{1}{3}$
4. $\frac{1}{1 + 2 e^{2}}$

Question 22 · Multiple choice

Let $f(x)=x^5+x$, and let $g$ be the inverse function of $f$. What is $g'(2)$?

1. $\frac{1}{6}$
2. $6$
3. $\frac{1}{81}$
4. $\frac{1}{2}$

Question 23 · Multiple choice

The function $g$ is the inverse of $f(x)=3x+\cos x$, which is increasing. What is $g'(1)$?

1. $3$
2. $\frac{1}{3 - \sin{\left(1 \right)}}$
3. $\frac{1}{3}$
4. $1$

Question 24 · Multiple choice

Let $f(x)=\sqrt{x+3}$ for $x\ge-3$, and let $g$ be the inverse function of $f$. What is $g'(2)$?

1. $\frac{1}{4}$
2. $2$
3. $4$
4. $\frac{1}{2}$

Question 25 · Multiple choice

If $f(x)=\arctan(2x)$, what is $f'(1)$?

1. $\frac{1}{5}$
2. $\frac{1}{2}$
3. $\frac{2}{5}$
4. $\frac{2}{3}$

Question 26 · Multiple choice

If $y=\arcsin\!\left(\dfrac{x}{2}\right)$, what is $\dfrac{dy}{dx}$ at $x=1$?

1. $\frac{2 \sqrt{3}}{3}$
2. $\frac{1}{2}$
3. $\frac{\sqrt{3}}{2}$
4. $\frac{\sqrt{3}}{3}$

Question 27 · Multiple choice

If $f(x)=x\arctan x$, what is $f'(1)$?

1. $\frac{1}{2} + \frac{\pi}{4}$
2. $\frac{\pi}{4}$
3. $\frac{1}{2}$
4. $\frac{\pi}{4} + 1$

Question 28 · Multiple choice

If $y=\arccos(x^2)$, what is $\dfrac{dy}{dx}$ at $x=\frac12$?

1. $- \frac{4 \sqrt{15}}{15}$
2. $\frac{4 \sqrt{15}}{15}$
3. $- \frac{2 \sqrt{3}}{3}$
4. $-1$

Question 29 · Multiple choice

What is the slope of the line tangent to $y=\arctan\sqrt{x}$ at $x=1$?

1. $\frac{1}{2}$
2. $\frac{\pi}{4}$
3. $\frac{1}{4}$
4. $1$

Question 30 · Multiple choice

If $g(x)=\arctan(e^x)$, what is $g'(0)$?

1. $1$
2. $\frac{\pi}{4}$
3. $\frac{1}{2}$
4. $2$

Question 31 · Multiple choice

If $f(x)=\dfrac{\ln(x^2+1)}{x}$, what is $f'(1)$?

1. $1 - \ln{\left(2 \right)}$
2. $-1 + \ln{\left(2 \right)}$
3. $1$
4. $\ln{\left(2 \right)} + 1$

Question 32 · Multiple choice

If $y=e^{3x}\sin x$, what is $\dfrac{dy}{dx}$ at $x=0$?

1. $3$
2. $0$
3. $1$
4. $4$

Question 33 · Multiple choice

If $f(x)=\dfrac{(x+1)^2}{x^2+3}$, what is $f'(1)$?

1. $\frac{1}{2}$
2. $1$
3. $\frac{1}{4}$
4. $- \frac{1}{2}$

Question 34 · Multiple choice

If $f(x)=\sin(\ln x)$, what is $f'(e^{\pi})$?

1. $- \frac{1}{e^{\pi}}$
2. $e^{- \pi}$
3. $-1$
4. $0$

Question 35 · Multiple choice

If $y=\tan^2(3x)$, what is $\dfrac{dy}{dx}$ at $x=\frac{\pi}{12}$?

1. $4$
2. $6$
3. $2$
4. $12$

Question 36 · Multiple choice

The table gives values of differentiable functions $f$ and $g$ and their derivatives. If $h(x)=\dfrac{f(x)}{g(2x)}$, what is $h'(1)$?

1. $- \frac{7}{8}$
2. $- \frac{11}{16}$
3. $- \frac{1}{2}$
4. $- \frac{5}{8}$

Question 37 · Multiple choice

Let $f(x)=\sin x\cdot\ln(x+2)$. What is the value of $f'(1)$?

1. $0.594$
2. $0.874$
3. $0.280$
4. $0.925$

Question 38 · Multiple choice

If $f(x)=x^4-3x^2$, what is $f''(1)$?

1. $-2$
2. $12$
3. $6$
4. $0$

Question 39 · Multiple choice

If $y=\sin(2x)$, what is $\dfrac{d^2y}{dx^2}$ at $x=\frac{\pi}{4}$?

1. $-4$
2. $4$
3. $-2$
4. $0$

Question 40 · Multiple choice

If $y=e^{-x^2}$, what is $\dfrac{d^2y}{dx^2}$ at $x=0$?

1. $-2$
2. $2$
3. $0$
4. $-1$

Question 41 · Multiple choice

For the circle $x^2+y^2=25$, what is $\dfrac{d^2y}{dx^2}$ at the point $(3,4)$?

1. $- \frac{3}{4}$
2. $\frac{25}{64}$
3. $- \frac{1}{4}$
4. $- \frac{25}{64}$

Question 42 · Multiple choice

If $f(x)=\ln x$, what is $f'''(1)$?

1. $-1$
2. $2$
3. $1$
4. $-2$

Question 43 · Free response (9 points)

Consider the curve given by $y^3+xy=12$. The point $(2,2)$ is on the curve.

1. (a) Show that $\dfrac{dy}{dx}=\dfrac{-y}{3y^2+x}$, and find the slope of the curve at $(2,2)$. [2 points]
2. (b) Write an equation for the line tangent to the curve at $(2,2)$, and use it to approximate the $y$-coordinate of the point on the curve near $(2,2)$ with $x=2.1$. [2 points]
3. (c) Find the value of $\dfrac{d^2y}{dx^2}$ at the point $(2,2)$. [3 points]
4. (d) Find the point on the curve at which the tangent line is vertical. [2 points]

Question 44 · Free response (9 points)

The functions $f$ and $g$ are differentiable for all real numbers, and $f$ is increasing. The table gives values of $f$, $f'$, $g$ and $g'$ at selected values of $x$.

1. (a) Let $h(x)=f\big(g(x)\big)$. Find $h'(2)$. [2 points]
2. (b) Let $f^{-1}$ be the inverse function of $f$. Find $\big(f^{-1}\big)'(6)$. [2 points]
3. (c) Let $m(x)=\arctan\big(g(x)\big)$. Find $m'(3)$. [2 points]
4. (d) Write an equation for the line tangent to the graph of $y=f^{-1}(x)$ at $x=4$. [3 points]