#AP Calculus AB/BC: Differentiating Inverse Functions

Hey there, future calculus master! 👋 Let's dive into differentiating inverse functions. This is a crucial skill, and we'll make sure you're totally confident with it by the end of this guide.

This topic is frequently tested on both Multiple Choice and Free Response Questions. Make sure you understand the core concept and practice different types of problems.

# 🔄 Differentiating Inverse Functions

#The Core Concept

If $f^{-1}(x)$ is the inverse of a differentiable and invertible function $f(x)$, then the derivative of the inverse function is given by:

rac{d}{dx}f^{-1}(x) = rac{1}{f'(f^{-1}(x))}

Or, if $g(x)$ is the inverse of $f(x)$:

rac{d}{dx}g(x) = rac{1}{f'(g(x))}

#Visualizing Inverse Functions

Graph created with Desmos

---

# 🧮 Practice Problems

Let's solidify your understanding with a couple of examples. We'll start with a straightforward calculation and then tackle a more complex problem using a table of values.

#1) Calculating the Inverse Function

Problem: If $f(x) = 2x + 1$, find $(f^{-1})'(1)$.

Solution:

1. Find $f^{-1}(x)$: Switch $x$ and $y$ and solve for $y$. $$x = 2f^{-1}(x) + 1$$ f^{-1}(x) = rac{x - 1}{2}

2. Evaluate $f^{-1}(1)$: f^{-1}(1) = rac{1 - 1}{2} = 0

3. Apply the Inverse Derivative Formula: rac{d}{dx}f^{-1}(1) = rac{1}{f'(f^{-1}(1))} = rac{1}{f'(0)}

4. Find $f'(x)$: $$f'(x) = 2$$ So, $f'(0) = 2$

5. Final Calculation: rac{d}{dx}f^{-1}(1) = rac{1}{2}

#2) Inverse Function Values from a Table

Problem: (Adapted from the 2007 AP Calculus AB exam) The functions $f$ and $g$ are differentiable for all real numbers, and $g$ is strictly increasing. The table below gives values of the functions and their first derivatives at selected values of $x$.

If $g^{-1}$ is the inverse function of $g$, write an equation for the line tangent to the graph of $y = g^{-1}(x)$ at $x = 2$.

#a. Find the slope of the tangent line at $x = 2$

1. Find $g^{-1}(2)$: We know $g(1) = 2$, so $g^{-1}(2) = 1$.

2. Apply the Inverse Derivative Formula: rac{d}{dx}g^{-1}(2) = rac{1}{g'(g^{-1}(2))} = rac{1}{g'(1)}

3. Use the table to find $g'(1) = 5$

4. Calculate the slope: rac{d}{dx}g^{-1}(2) = rac{1}{5}

#b. Write the equation of the tangent line

1. Use the point-slope form: $y - y_1 = m(x - x_1)$

2. Plug in the values: m = rac{1}{5}, $x_1 = 2$, and $y_1 = g^{-1}(2) = 1$

3. Tangent line equation: y - 1 = rac{1}{5}(x - 2)

# 🌟 Closing

Fantastic job working through these problems! You're now equipped to tackle differentiating inverse functions. Remember, practice is key, so keep reviewing and working on more examples.

Image Courtesy of Giphy

# 🎯 Final Exam Focus

#High-Priority Topics

• Inverse Function Derivative Formula: Know it inside and out! It's the foundation for these types of problems.
• Table Values: Be ready to use tables to find function and derivative values.
• Tangent Lines: Understand how to combine inverse derivatives with tangent line equations.

#Common Question Types

• Multiple Choice: Expect to see direct applications of the inverse derivative formula.
• Free Response: Be prepared for multi-part problems involving tables, graphs, and tangent lines.

#Last-Minute Tips

• Time Management: Don't spend too long on one problem. If you're stuck, move on and come back later.
• Careful Calculations: Double-check your work, especially when dealing with fractions and negative signs.
• Stay Calm: You've got this! Take deep breaths and approach each problem with confidence.

# 📝 Practice Questions

Practice Question

#Multiple Choice Questions

1. If $f(x) = x^3 + 2x - 1$, and $g(x)$ is the inverse of $f(x)$, what is $g'(2)$? (A) 1/14 (B) 1/5 (C) 1/3 (D) 5

2. Given the function $h(x)$ and its inverse $h^{-1}(x)$, and $h(3) = 5$ and $h'(3) = 2$, what is $(h^{-1})'(5)$? (A) 1/5 (B) 1/2 (C) 2 (D) 5

#Free Response Question

Let $f(x)$ be a differentiable function with selected values given in the table below:

x	1	2	3	4
f(x)	2	3	5	8
f'(x)	1	2	3	4

Let $g(x)$ be the inverse of $f(x)$.

(a) Find $g'(3)$.

(b) Write an equation for the line tangent to the graph of $g(x)$ at $x = 3$.

(c) If $h(x) = (g(x))^2$, find $h'(3)$.

Scoring Breakdown:

(a) 2 points

• 1 point for correctly identifying $g(3) = 2$
• 1 point for correct answer: $g'(3) = 1/f'(g(3)) = 1/f'(2) = 1/2$

(b) 3 points

• 1 point for correct slope from part (a)
• 1 point for correct point (3, 2)
• 1 point for correct tangent line equation: $y - 2 = (1/2)(x - 3)$

(c) 4 points

• 1 point for correctly applying chain rule: $h'(x) = 2g(x)g'(x)$
• 1 point for correct $g(3) = 2$
• 1 point for correct $g'(3) = 1/2$
• 1 point for correct answer: $h'(3) = 2(2)(1/2) = 2$

Keep up the great work! You're on your way to acing this exam! 🚀