AP Calculus AB Derivatives of Inverse Functions Lesson (Free PDF)

AP Calculus AB math lesson: Derivatives of Inverse Functions – Maths PDF

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This inverse derivatives lesson 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.

The derivative of an inverse function is the reciprocal of the original derivative: (f⁻¹)′(b) = 1/f′(a), where f(a) = b. The derivatives of the inverse trig functions are 1/√(1-x²) for sin⁻¹x, -1/√(1-x²) for cos⁻¹x and 1/(1+x²) for tan⁻¹x. Higher-order derivatives repeat differentiation.

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

Key facts in this inverse derivatives lesson

Key facts

(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 do you differentiate an inverse function?

If \(f\) has an inverse \(f^{-1}\), then the graphs are mirror images across the line \(y=x\). Reflection swaps the roles of rise and run, so the slopes are reciprocals.

Derivative of an inverse

\(\left(f^{-1}\right)^{\prime}(b)=\frac{1}{f^{\prime}(a)}\) where \(f(a)=b\), provided \(f^{\prime}(a)\neq 0\). Evaluate \(f^{\prime}\) at the input of the original function.

Reflecting across y = x swaps the points (2, 8) and (8, 2) and turns the slope 12 into 1/12.

Example. For \(f(x)=x^{3}\), \(f(2)=8\) and \(f^{\prime}(2)=12\). So \(\left(f^{-1}\right)^{\prime}(8)=\frac{1}{12}\).

What are the derivatives of inverse trig functions?

Function	Derivative
\(\sin^{-1}x\)	\(\frac{1}{\sqrt{1-x^{2}}}\)
\(\cos^{-1}x\)	\(-\frac{1}{\sqrt{1-x^{2}}}\)
\(\tan^{-1}x\)	\(\frac{1}{1+x^{2}}\)

When the input is more than \(x\), add the chain rule. For \(\tan^{-1}(x^{2})\), the derivative is \(\frac{2x}{1+x^{4}}\).

The derivative 1/(1 + x²) (green) is the slope of arctan x (blue).

What are higher-order derivatives?

The second derivative \(f^{\prime\prime}\) is the derivative of \(f^{\prime}\). It is also written \(\frac{d^{2}y}{dx^{2}}\). In motion problems, if \(s(t)\) is position, then \(v=s^{\prime}\) is velocity and \(a=s^{\prime\prime}\) is acceleration.

Each derivative is the rate of change of the one before it.

Example. For \(f(x)=x^{4}-3x^{3}+2x\): \(f^{\prime}=4x^{3}-9x^{2}+2\), \(f^{\prime\prime}=12x^{2}-18x\) and \(f^{\prime\prime\prime}=24x-18\).

Example with a table. If \(f(1)=7\) and \(f^{\prime}(1)=3\), then the inverse passes through \((7,1)\) with slope \(\frac{1}{3}\). Be careful about the order: the input of \(f^{-1}\) is \(7\), but the derivative of \(f\) is evaluated at \(1\). Many errors come from using \(f^{\prime}(7)\) by mistake.

Higher-order derivatives also reveal patterns. For \(e^{3x}\), each derivative multiplies by \(3\), so the fourth derivative is \(81e^{3x}\). For \(\sin x\), the derivatives repeat every four steps: \(\cos x\), \(-\sin x\), \(-\cos x\), \(\sin x\).

How do you choose a method?

Read the outermost operation first. A sum is differentiated term by term. A product needs the product rule, a quotient needs the quotient rule, and a function inside another function needs the chain rule. If \(y\) appears in an equation, use implicit differentiation.

A quick guide for choosing a differentiation procedure.

Inverse derivatives lesson: questions and answers

How do you find the derivative of an inverse function?

Find the input a with f(a) = b, compute f′(a), and take its reciprocal. The result is (f inverse)′(b) = 1/f′(a).

Why are the slopes of f and its inverse reciprocals?

The graphs are reflections across y = x, which swaps rise and run. A tangent line with slope m becomes a tangent line with slope 1/m.

What is the derivative of arctan x?

The derivative of arctan x is 1/(1 + x^2). It is always positive, which matches the increasing graph of arctan.

What is the second derivative?

The second derivative is the derivative of the first derivative. It measures how fast the slope is changing, and in motion it is the acceleration.

How do you decide which differentiation rule to use?

Look at the outermost operation. Sums use term-by-term differentiation, products and quotients use their own rules, nested functions use the chain rule, and equations mixing x and y use implicit differentiation.

How to use this inverse derivatives lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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