AP Calculus AB Product Rule, Quotient Rule and Trig Derivatives Worksheet Solutions (PDF)

AP Calculus AB math worksheet solutions: Product Rule, Quotient Rule and Trig Derivatives – Maths PDF

Cover of the book AP Calculus AB Math Worksheets with Answers (PDF)

Download as a PDFThe AP Calculus AB worksheets book with answersPrintable PDFSee the book ›

Sommaire

This product quotient answers for AP Calculus AB covers product rule, quotient rule and trig derivatives (Common Core AP CALC AB 2.8, AP CALC AB 2.9, AP CALC AB 2.10), with a free PDF to print.

This answer key gives a full worked solution for every problem in the product rule worksheet. Compare each step, not only the final answer, because AP graders award credit for a correct setup and clear justification. Final answers are in bold. Redo any problem you missed before moving on to the test.

AP Calculus AB · Common Core State Standards: AP CALC AB 2.8, AP CALC AB 2.9, AP CALC AB 2.10

Product quotient answers: exercises 1 to 10

Exercise 1 – Product with an exponential

Let $u=x^{2}$ and $v=e^{x}$. Then $u^{\prime}=2 x$ and $v^{\prime}=e^{x}$.

Product rule: $(uv)^{\prime}=u^{\prime}v+uv^{\prime}=(2 x)(e^{x})+(x^{2})(e^{x})$.

Final answer: $f^{\prime}(x)=\left(x^{2} + 2 x\right) e^{x}$.

Exercise 2 – Product of two polynomials

Let $u=3 x + 1$ and $v=x^{2} – 4$. Then $u^{\prime}=3$ and $v^{\prime}=2 x$.

Product rule: $(uv)^{\prime}=u^{\prime}v+uv^{\prime}=(3)(x^{2} – 4)+(3 x + 1)(2 x)$.

Final answer: $f^{\prime}(x)=9 x^{2} + 2 x – 12$.

Check: $(3x+1)(x^{2}-4)=3x^{3}+x^{2}-12x-4$, whose derivative is $9x^{2}+2x-12$. The answers match.

Exercise 3 – Product with sine

Let $u=x$ and $v=\sin{\left(x \right)}$. Then $u^{\prime}=1$ and $v^{\prime}=\cos{\left(x \right)}$.

Product rule: $(uv)^{\prime}=u^{\prime}v+uv^{\prime}=(1)(\sin{\left(x \right)})+(x)(\cos{\left(x \right)})$.

Final answer: $f^{\prime}(x)=x \cos{\left(x \right)} + \sin{\left(x \right)}$.

The derivative f′(x) = sin x + x cos x.

Exercise 4 – Exponential times log

Let $u=e^{x}$ and $v=\ln{\left(x \right)}$. Then $u^{\prime}=e^{x}$ and $v^{\prime}=\frac{1}{x}$.

Product rule: $(uv)^{\prime}=u^{\prime}v+uv^{\prime}=(e^{x})(\ln{\left(x \right)})+(e^{x})(\frac{1}{x})$.

Final answer: $f^{\prime}(x)=e^{x} \ln{\left(x \right)} + \frac{e^{x}}{x}$.

Exercise 5 – Square root times cosine

Let $u=\sqrt{x}$ and $v=\cos{\left(x \right)}$. Then $u^{\prime}=\frac{1}{2 \sqrt{x}}$ and $v^{\prime}=- \sin{\left(x \right)}$.

Product rule: $(uv)^{\prime}=u^{\prime}v+uv^{\prime}=(\frac{1}{2 \sqrt{x}})(\cos{\left(x \right)})+(\sqrt{x})(- \sin{\left(x \right)})$.

Final answer: $f^{\prime}(x)=- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$.

Exercise 6 – Simple quotient

Let $u=x$ and $v=x + 1$. Then $u^{\prime}=1$ and $v^{\prime}=1$.

Quotient rule: $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}=\frac{(1)(x + 1)-(x)(1)}{(x + 1)^{2}}$.

Final answer: $f^{\prime}(x)=\frac{1}{\left(x + 1\right)^{2}}$.

Exercise 7 – Quotient with a quadratic

Let $u=x^{2} + 1$ and $v=x – 3$. Then $u^{\prime}=2 x$ and $v^{\prime}=1$.

Quotient rule: $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}=\frac{(2 x)(x – 3)-(x^{2} + 1)(1)}{(x – 3)^{2}}$.

Final answer: $f^{\prime}(x)=\frac{x^{2} – 6 x – 1}{\left(x – 3\right)^{2}}$.

Exercise 8 – Cosine over a power

Let $u=\cos x$ and $v=x^{2}$. Then $u^{\prime}=-\sin x$ and $v^{\prime}=2x$.

Quotient rule: $\frac{-\sin x\cdot x^{2}-\cos x\cdot 2x}{x^{4}}=\frac{-x\sin x-2\cos x}{x^{3}}$.

Final answer: $f^{\prime}(x)=-\frac{x\sin x+2\cos x}{x^{3}}$.

Exercise 9 – Sine over x

Let $u=\sin{\left(x \right)}$ and $v=x$. Then $u^{\prime}=\cos{\left(x \right)}$ and $v^{\prime}=1$.

Quotient rule: $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}=\frac{(\cos{\left(x \right)})(x)-(\sin{\left(x \right)})(1)}{(x)^{2}}$.

Final answer: $f^{\prime}(x)=\frac{x \cos{\left(x \right)} – \sin{\left(x \right)}}{x^{2}}$.

Exercise 10 – Log over a power

Let $u=\ln{\left(x \right)}$ and $v=x^{3}$. Then $u^{\prime}=\frac{1}{x}$ and $v^{\prime}=3 x^{2}$.

Quotient rule: $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}=\frac{(\frac{1}{x})(x^{3})-(\ln{\left(x \right)})(3 x^{2})}{(x^{3})^{2}}$.

Final answer: $f^{\prime}(x)=\frac{1 – 3 \ln{\left(x \right)}}{x^{4}}$.

Exercises 11 and up

Exercise 11 – Derivative of tangent

$\frac{d}{dx}[\tan x]=\sec^{2}x$.

At $x=\frac{\pi}{4}$, $\sec\frac{\pi}{4}=\sqrt{2}$, so $\sec^{2}\frac{\pi}{4}=2$.

Final answer: $\sec^{2}x$ and the value $2$.

The tangent to tan x at x = π/4 has slope 2.

Exercise 12 – Product with tangent

Let $u=x$ and $v=\tan{\left(x \right)}$. Then $u^{\prime}=1$ and $v^{\prime}=\tan^{2}{\left(x \right)} + 1$.

Product rule: $(uv)^{\prime}=u^{\prime}v+uv^{\prime}=(1)(\tan{\left(x \right)})+(x)(\tan^{2}{\left(x \right)} + 1)$.

Final answer: $f^{\prime}(x)=x \sec^{2}{\left(x \right)} + \tan{\left(x \right)}$.

Exercise 13 – Secant at a point

$\frac{d}{dx}[\sec x]=\sec x\tan x$. At $\frac{\pi}{3}$: $\sec\frac{\pi}{3}=2$ and $\tan\frac{\pi}{3}=\sqrt{3}$.

Final answer: $2\sqrt{3}$.

Exercise 14 – Cotangent and cosecant

$f^{\prime}(x)=-\csc^{2}x-\csc x\cot x$.

Final answer: $-\csc x(\csc x+\cot x)$.

Exercise 15 – Tangent line with a product

$f(\pi)=\pi^{2}\sin\pi=0$. $f^{\prime}(x)=2x\sin x+x^{2}\cos x$, so $f^{\prime}(\pi)=0+\pi^{2}(-1)=-\pi^{2}$.

Tangent: $y-0=-\pi^{2}(x-\pi)$. Final answer: $y=-\pi^{2}(x-\pi)$.

Exercise 16 – Horizontal tangent

$f^{\prime}(x)=e^{x}+xe^{x}=e^{x}(1+x)$. Since $e^{x} > 0$, set $1+x=0$.

Final answer: $x=-1$, the point $\left(-1,-\frac{1}{e}\right)$.

Exercise 17 – Quotient and a horizontal tangent

Let $u=x^{2}$ and $v=x^{2} + 1$. Then $u^{\prime}=2 x$ and $v^{\prime}=2 x$.

Quotient rule: $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}=\frac{(2 x)(x^{2} + 1)-(x^{2})(2 x)}{(x^{2} + 1)^{2}}$.

Final answer: $f^{\prime}(x)=\frac{2 x}{\left(x^{2} + 1\right)^{2}}$.

The numerator is zero only at $x=0$, so the only horizontal tangent is at $(0,0)$.

Exercise 18 – Using a table of values

Product: $f^{\prime}g+fg^{\prime}=(-1)(4)+(3)(5)=11$.

Quotient: $\frac{f^{\prime}g-fg^{\prime}}{g^{2}}=\frac{(-1)(4)-(3)(5)}{16}=-\frac{19}{16}$.

Final answer: $11$ and $-\frac{19}{16}$.

Exercise 19 – Revenue growth

$p(4)=12$, $q(4)=184$, $p^{\prime}=0.5$, $q^{\prime}=-4$.

Product rule: $R^{\prime}(4)=p^{\prime}q+pq^{\prime}=0.5(184)+12(-4)=92-48=44$.

Final answer: \$44 per week: at 4 weeks the revenue is growing by about \$44 each week.

Revenue in dollars during the first 20 weeks.

Exercise 20 – Peak concentration

Let $u=x$ and $v=x^{2} + 4$. Then $u^{\prime}=1$ and $v^{\prime}=2 x$.

Quotient rule: $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}=\frac{(1)(x^{2} + 4)-(x)(2 x)}{(x^{2} + 4)^{2}}$.

Final answer: $f^{\prime}(x)=\frac{4 – x^{2}}{\left(x^{2} + 4\right)^{2}}$.

Set $4-t^{2}=0$: $t=2$ (hours cannot be negative).

$C(2)=\frac{2}{8}=0.25$. The concentration is greatest at 2 hours (0.25 mg/L).

Exercise 21 – Three factors

Apply the product rule twice. Group $x^{2}e^{x}$ as the first factor, whose derivative is $e^{x}(x^{2}+2x)$.

$f^{\prime}(x)=e^{x}(x^{2}+2x)\sin x+x^{2}e^{x}\cos x$.

Final answer: $xe^{x}[(x+2)\sin x+x\cos x]$.

Exercise 22 – Proving the tangent derivative

$u=\sin x$, $v=\cos x$, $u^{\prime}=\cos x$, $v^{\prime}=-\sin x$.

$\frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^{2}x}=\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}=\frac{1}{\cos^{2}x}=\sec^{2}x$.

Final answer: $(\tan x)^{\prime}=\sec^{2}x$.

Key facts behind this product quotient answers

Key facts

Product rule: (uv)′ = u′v + uv′.
Quotient rule: (u/v)′ = (u′v – uv′)/v², and the order in the numerator matters.
d/dx tan x = sec²x and d/dx sec x = sec x tan x.
The derivatives of cot x and csc x are negative: -csc²x and -csc x cot x.

How to use this product quotient answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

Download the product quotient answers as a PDF with the button below and keep it with your notes.

All AP Calculus AB chapters: AP Calculus AB math.

Voter.. post

Free printable PDF