AP Calculus AB Derivatives of Inverse Functions Worksheet Solutions (PDF)
This inverse derivatives answers for AP Calculus AB covers derivatives of inverse functions (Common Core AP CALC AB 3.3, AP CALC AB 3.4, AP CALC AB 3.5, AP CALC AB 3.6), with a free PDF to print.
This answer key gives a full worked solution for every problem in the inverse derivatives 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 3.3, AP CALC AB 3.4, AP CALC AB 3.5, AP CALC AB 3.6
Inverse derivatives answers: exercises 1 to 10
Exercise 1 – Inverse derivative from values
Since \(f(2)=5\), \(f^{-1}(5)=2\). Then \(\left(f^{-1}\right)^{\prime}(5)=\frac{1}{f^{\prime}(2)}=\frac{1}{4}\).
Final answer: \(\frac{1}{4}\).
Exercise 2 – Inverse of a cubic
Find \(a\) with \(f(a)=3\): \(a=1\) since \(1+1+1=3\).
\(f^{\prime}(x)=3x^{2}+1\), so \(f^{\prime}(1)=4\).
\(\left(f^{-1}\right)^{\prime}(3)=\frac{1}{4}\).
Final answer: \(\frac{1}{4}\).
Exercise 3 – Inverse with sine
\(f(0)=0\), so \(a=0\). \(f^{\prime}(x)=2+\cos x\), so \(f^{\prime}(0)=3\).
\(\left(f^{-1}\right)^{\prime}(0)=\frac{1}{3}\).
Final answer: \(\frac{1}{3}\).
Exercise 4 – Inverse of the exponential
Here \(f(2)=e^{2}\) and \(f^{\prime}(2)=e^{2}\). So \(\left(f^{-1}\right)^{\prime}(e^{2})=\frac{1}{e^{2}}\).
This agrees with \(\frac{d}{dx}[\ln x]=\frac{1}{x}\) at \(x=e^{2}\).
Final answer: \(\frac{1}{e^{2}}\).
Exercise 5 – Inverse of a fifth-degree polynomial
\(f(1)=2\), so \(a=1\). \(f^{\prime}(x)=5x^{4}+1\) gives \(f^{\prime}(1)=6\).
Final answer: \(\left(f^{-1}\right)^{\prime}(2)=\frac{1}{6}\).
Exercise 6 – Inverse from a table
\(f(1)=7\), so \(\left(f^{-1}\right)^{\prime}(7)=\frac{1}{f^{\prime}(1)}=\frac{1}{3}\).
\(f(3)=12\), so \(\left(f^{-1}\right)^{\prime}(12)=\frac{1}{f^{\prime}(3)}=\frac{1}{6}\).
Final answer: \(\frac{1}{3}\text{ and }\frac{1}{6}\).
Exercise 7 – Arcsine at a point
\(\frac{1}{\sqrt{1-x^{2}}}\). At \(x=\frac{1}{2}\): \(\frac{1}{\sqrt{1-\frac{1}{4}}}=\frac{1}{\sqrt{3}/2}=\frac{2}{\sqrt{3}}\).
Final answer: \(\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\).
Exercise 8 – Arctangent at a point
\(\frac{dy}{dx}=\frac{1}{1+x^{2}}\), so at \(x=1\) the slope is \(\frac{1}{1+1}=\frac{1}{2}\).
Final answer: \(\frac{1}{2}\).
Exercise 9 – Arcsine with a chain
Outer \(\sin^{-1}u\), inner \(3x\): \(\frac{1}{\sqrt{1-(3x)^{2}}}\cdot 3\).
Final answer: \(f^{\prime}(x)=\frac{3}{\sqrt{1-9x^{2}}}\).
Exercise 10 – Arctangent of a square
\(\frac{1}{1+(x^{2})^{2}}\cdot 2x\).
Final answer: \(f^{\prime}(x)=\frac{2x}{1+x^{4}}\).
Exercises 11 and up
Exercise 11 – Arccosine of x over 2
\(-\frac{1}{\sqrt{1-\frac{x^{2}}{4}}}\cdot\frac{1}{2}=-\frac{1}{2\sqrt{\frac{4-x^{2}}{4}}}=-\frac{1}{\sqrt{4-x^{2}}}\).
Final answer: \(f^{\prime}(x)=-\frac{1}{\sqrt{4-x^{2}}}\).
Exercise 12 – Product with arctangent
Product rule: \(1\cdot\tan^{-1}x+x\cdot\frac{1}{1+x^{2}}\).
Final answer: \(f^{\prime}(x)=\tan^{-1}x+\frac{x}{1+x^{2}}\).
Exercise 13 – Exponential times arcsine
Product rule: \(e^{x}\sin^{-1}x+e^{x}\cdot\frac{1}{\sqrt{1-x^{2}}}\).
Final answer: \(f^{\prime}(x)=e^{x}\left(\sin^{-1}x+\frac{1}{\sqrt{1-x^{2}}}\right)\).
Exercise 14 – Arctangent of an exponential
\(\frac{1}{1+(e^{x})^{2}}\cdot e^{x}\).
Final answer: \(f^{\prime}(x)=\frac{e^{x}}{1+e^{2x}}\).
Exercise 15 – Three derivatives
\(f^{\prime}(x)=4x^{3}-9x^{2}+2\). \(f^{\prime\prime}(x)=12x^{2}-18x\). \(f^{\prime\prime\prime}(x)=24x-18\).
\(f^{\prime\prime}(1)=12-18=-6\).
Final answer: \(f^{\prime\prime}(1)=-6\).
Exercise 16 – Second derivative of x e to the x
Product rule: \(f^{\prime}=e^{x}+xe^{x}=(x+1)e^{x}\).
Again: \(f^{\prime\prime}=e^{x}+(x+1)e^{x}=(x+2)e^{x}\).
Final answer: \(f^{\prime\prime}(x)=(x+2)e^{x}\).
Exercise 17 – A pattern of derivatives
\(f^{\prime}=3e^{3x}\), \(f^{\prime\prime}=9e^{3x}\), \(f^{\prime\prime\prime}=27e^{3x}\). Each derivative multiplies by \(3\).
Final answer: \(f^{(4)}(x)=81e^{3x}\).
Exercise 18 – Second derivative of a log
\(f^{\prime}=x^{-1}\), so \(f^{\prime\prime}=-x^{-2}\).
Final answer: \(-\frac{1}{x^{2}}\).
Exercise 19 – Velocity and acceleration
\(v(t)=3t^{2}-18t+15=3(t-1)(t-5)\) and \(a(t)=6t-18\).
At rest: \(t=1\) and \(t=5\). Accelerations: \(a(1)=-12\) and \(a(5)=12\) ft/s².
Final answer: \(t=1,\ 5;\ a(1)=-12,\ a(5)=12\text{ ft/s}^{2}\).
Exercise 20 – Choosing the rule
- Product rule: \(2x\sin x+x^{2}\cos x\).
- Quotient rule: \(\frac{2x(x-1)-(x^{2}+1)}{(x-1)^{2}}=\frac{x^{2}-2x-1}{(x-1)^{2}}\).
- Chain rule: \(\frac{6x}{2\sqrt{3x^{2}+1}}=\frac{3x}{\sqrt{3x^{2}+1}}\).
- Term by term: \(3x^{2}-5\).
Exercise 21 – Implicit second derivative
\(y+xy^{\prime}=0\), so \(y^{\prime}=-\frac{y}{x}\).
Quotient rule: \(y^{\prime\prime}=-\frac{y^{\prime}x-y}{x^{2}}=-\frac{-y-y}{x^{2}}=\frac{2y}{x^{2}}\).
With \(y=\frac{1}{x}\): \(\frac{2}{x^{3}}\). Check: \(\frac{d^{2}}{dx^{2}}[x^{-1}]=2x^{-3}\). They match.
Final answer: \(y^{\prime\prime}=\frac{2y}{x^{2}}=\frac{2}{x^{3}}\).
Exercise 22 – Volume and radius
\(V(3)=\frac{4}{3}\pi\cdot 27=36\pi\), so the radius is \(3\) inches. \(V^{\prime}(r)=4\pi r^{2}\), so \(V^{\prime}(3)=36\pi\).
\(r^{\prime}(36\pi)=\frac{1}{V^{\prime}(3)}=\frac{1}{36\pi}\).
Final answer: \(\frac{1}{36\pi}\approx 0.0088\text{ in/in}^{3}\): when the balloon holds \(36\pi\) cubic inches, each extra cubic inch of air adds about 0.0088 inch of radius.
Exercise 23 – Temperature scales
\(F^{\prime}(C)=1.8\) for every \(C\). The inverse derivative is \(\frac{1}{1.8}=\frac{5}{9}\).
Final answer: \(\frac{5}{9}\).
A change of 1 °F corresponds to a change of \(\frac{5}{9}\) °C.
Key facts behind this inverse derivatives answers
- (f⁻¹)′(b) = 1/f′(a) where f(a) = b; evaluate f′ at a, not at b.
- d/dx sin⁻¹x = 1/√(1-x²), d/dx cos⁻¹x = -1/√(1-x²), d/dx tan⁻¹x = 1/(1+x²).
- The second derivative is the derivative of the first derivative; acceleration is the second derivative of position.
- Choose the rule by the outermost operation: sum, product, quotient or composition.
How to use this inverse derivatives 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.
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Download the inverse derivatives answers as a PDF with the button below and keep it with your notes.
Download the inverse derivatives answers as a PDF with the button below and keep it with your notes.
Download the inverse derivatives answers as a PDF with the button below and keep it with your notes.
Download the inverse derivatives answers as a PDF with the button below and keep it with your notes.
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