Differentiating Inverse Functions Chapter Notes | Calculus AB - Grade 9 PDF Download

Welcome to our AP Calculus study session! Today, we’re diving into differentiating inverse functions, building on our knowledge of the chain rule and implicit differentiation. Let’s strengthen those derivative skills!

Understanding Inverse Function Derivatives

Practice Problems

Problem 1: Finding the Derivative of an Inverse Function

Problem 2: Tangent Line to an Inverse Function (2007 AP Calculus AB FRQ)

If g⁻¹ is the inverse of g, find the equation of the tangent line to y = g⁻¹(x) at x = 2.
Step a: Find the slope at x = 2
We need the slope of the tangent to y = g⁻¹(x) at x = 2. Since g(1) = 2, it follows that g⁻¹(2) = 1. Using the inverse derivative formula:
d/dx [g⁻¹(2)] = 1 / g'(g⁻¹(2)) = 1 / g'(1)
From the table, g'(1) = 5, so:
d/dx [g⁻¹(2)] = 1 / 5
Step b: Write the tangent line equation
To write the equation of the line tangent to the graph of y = g⁻¹(x) at x = 2, we will need to plug into this formula: 
y − y₁ = m(x − x₁).
We just calculated the value of mm, which equals . We know that x₁ = 2, and found that y(2) = g⁻¹(2) = 1 
We can just plug it into the equation now! We can now answer that y − 1 = 1/5(x−2)
Amazing job! Our answers would have earned 3/3 points for this question. This one was tough since we had three different functions to look at: g(x), g⁻¹(x), and g′(x), and had to take the calculations a step further.