AP Calculus AB Derivatives of Inverse Functions Worksheet (PDF)
This inverse derivatives worksheet 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.
These 23 problems practice derivatives of inverse functions, derivatives of inverse trigonometric functions, choosing differentiation methods and higher-order derivatives (AP Calculus AB, topics 3.3, 3.4, 3.5, 3.6). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.
Later problems cover tables of values, products with inverse trig functions, second derivatives and situations such as a balloon volume and a cart on a track. Work in pencil, show every step, and check the answer key only after a full attempt.
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 worksheet: exercises 1 to 10
Exercise 1 – Inverse derivative from values
Exercise 2 – Inverse of a cubic
Let \(f(x)=x^{3}+x+1\). Find \(\left(f^{-1}\right)^{\prime}(3)\).
Exercise 3 – Inverse with sine
Exercise 4 – Inverse of the exponential
Let \(f(x)=e^{x}\). Find \(\left(f^{-1}\right)^{\prime}(e^{2})\) and compare with \(\frac{d}{dx}[\ln x]\).
Exercise 5 – Inverse of a fifth-degree polynomial
Exercise 6 – Inverse from a table
Use the table to find \(\left(f^{-1}\right)^{\prime}(7)\) and \(\left(f^{-1}\right)^{\prime}(12)\).
Exercise 7 – Arcsine at a point
Exercise 8 – Arctangent at a point
Find the slope of \(y=\tan^{-1}x\) at \(x=1\).
Exercise 9 – Arcsine with a chain
Exercise 10 – Arctangent of a square
Exercises 11 and up
Exercise 11 – Arccosine of x over 2
Exercise 12 – Product with arctangent
Exercise 13 – Exponential times arcsine
Exercise 14 – Arctangent of an exponential
Exercise 15 – Three derivatives
Exercise 16 – Second derivative of x e to the x
Exercise 17 – A pattern of derivatives
Exercise 18 – Second derivative of a log
Exercise 19 – Velocity and acceleration
A cart on a track has position \(s(t)=t^{3}-9t^{2}+15t\) feet after \(t\) seconds. Find \(v(t)\), \(a(t)\), the times when the cart is at rest, and the acceleration at those times.
Exercise 20 – Choosing the rule
Exercise 21 – Implicit second derivative
Exercise 22 – Volume and radius
The volume of a balloon in cubic inches is \(V(r)=\frac{4}{3}\pi r^{3}\). Let \(r(V)\) be the inverse function. Find \(r^{\prime}(36\pi)\) and interpret it.
Exercise 23 – Temperature scales
Key facts behind this inverse derivatives worksheet
- (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 worksheet
Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.
Download the inverse derivatives worksheet as a PDF with the button below and keep it with your notes.
Download the inverse derivatives worksheet as a PDF with the button below and keep it with your notes.
All AP Calculus AB chapters: AP Calculus AB math.
