Derivatives of Inverse Functions

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Key Ideas Easy Medium Hard

\(\bigl(f^{-1}\bigr)'(a) = \dfrac{1}{f'(b)}\ \text{where}\ f(b) = a\)

What to Remember

Flip the point, then take the reciprocal of the slope

If \(f(b) = a\) and \(f'(b) \neq 0\),

\(\bigl(f^{-1}\bigr)'(a) = \dfrac{1}{f'(b)}\ \text{where}\ f(b) = a\)

The inverse sends \(a\) back to \(b\).

A zero derivative on f makes the inverse derivative fail

If \(f'(b) = 0\), then \(\bigl(f^{-1}\bigr)'\) does not exist at \(f(b)\). The inverse has a vertical tangent there.

Find b before you differentiate

Solve \(f(b) = a\) first. Then compute \(f'(b)\). The reciprocal is the last line.

The chain rule still wraps an inverse

For \(h(x) = f^{-1}\!\bigl(g(x)\bigr)\),

\(h'(x) = \bigl(f^{-1}\bigr)'\!\bigl(g(x)\bigr)\cdot g'(x)\)

Level 1

Easy

The table or a short equation tells you b and f'(b).

Example

Use the table. f is one-to-one. Find \(\bigl(f^{-1}\bigr)'(3)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

\(\bigl(f^{-1}\bigr)'(a) = \dfrac{1}{f'(b)}\ \text{where}\ f(b) = a\)

\(f(1) = 3\), so \(b = 1\).

\(f'(1) = 4\)

\(\dfrac{1}{4}\)

Answer: \(\dfrac{1}{4}\)

Use the table. Find \(\bigl(f^{-1}\bigr)'(5)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

Show Answer

\(f(2) = 5\)

\(f'(2) = -2\)

\(\dfrac{1}{-2}\)

\(-\dfrac{1}{2}\)

Answer: \(-\dfrac{1}{2}\)

Use the table. Find \(\bigl(f^{-1}\bigr)'(8)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

Show Answer

\(f(3) = 8\)

\(f'(3) = 2\)

\(\dfrac{1}{2}\)

Answer: \(\dfrac{1}{2}\)

Use the table. Find \(\bigl(f^{-1}\bigr)'(2)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

Show Answer

\(f(4) = 2\)

\(f'(4) = 3\)

\(\dfrac{1}{3}\)

Answer: \(\dfrac{1}{3}\)

Let \(f(x) = x^3 + x\). Find \(\bigl(f^{-1}\bigr)'(2)\).

Show Answer

\(f(1) = 1 + 1 = 2\)

\(f'(x) = 3x^2 + 1\)

\(f'(1) = 4\)

\(\dfrac{1}{4}\)

Answer: \(\dfrac{1}{4}\)

Let \(f(x) = x^5 + 2x + 1\). Find \(\bigl(f^{-1}\bigr)'(1)\).

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\(f(0) = 1\)

\(f'(x) = 5x^4 + 2\)

\(f'(0) = 2\)

\(\dfrac{1}{2}\)

Answer: \(\dfrac{1}{2}\)

Level 2

Medium

A graph, a chain around the inverse, or a derivative that is zero.

Example

Let \(f(x) = e^x\). Find \(\bigl(f^{-1}\bigr)'\!\bigl(e^2\bigr)\).

\(f(2) = e^2\)

\(f'(x) = e^x\)

\(f'(2) = e^2\)

\(\dfrac{1}{e^2}\)

Answer: \(e^{-2}\)

The graph of f is the line through \((1, 3)\) with slope 4. Find \(\bigl(f^{-1}\bigr)'(3)\).

Graph of f

Show Answer

The closed point is \((1, 3)\), so \(f(1) = 3\).

\(f'(1) = 4\)

\(\bigl(f^{-1}\bigr)'(3) = \dfrac{1}{4}\)

Answer: \(\dfrac{1}{4}\)

Use the table. Let \(h(x) = f^{-1}(2x)\). Find \(h'\!\bigl(\tfrac{3}{2}\bigr)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

Show Answer

\(h'(x) = \bigl(f^{-1}\bigr)'(2x)\cdot 2\)

Substitute \(x = \tfrac{3}{2}\).

\(2x = 3\)

\(\bigl(f^{-1}\bigr)'(3) = \dfrac{1}{4}\)

\(\dfrac{1}{4}\cdot 2\)

\(\dfrac{1}{2}\)

Answer: \(\dfrac{1}{2}\)

Let \(f(x) = x^3 + 2x\). Find \(\bigl(f^{-1}\bigr)'(3)\).

Show Answer

\(f(1) = 3\)

\(f'(x) = 3x^2 + 2\)

\(f'(1) = 5\)

\(\dfrac{1}{5}\)

Answer: \(\dfrac{1}{5}\)

Let \(f(x) = x^3\). Does \(\bigl(f^{-1}\bigr)'(0)\) exist?

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\(f(0) = 0\)

\(f'(x) = 3x^2\)

\(f'(0) = 0\)

The reciprocal of 0 is undefined.

The inverse has a vertical tangent at 0.

Answer: does not exist

Use the table. Which value equals \(\bigl(f^{-1}\bigr)'(1)\)?

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

A. \(\dfrac{1}{6}\)

B. \(6\)

C. \(-\dfrac{1}{2}\)

D. \(\dfrac{1}{4}\)

Show Answer

\(f(5) = 1\)

\(f'(5) = 6\)

\(\dfrac{1}{6}\)

Answer: A. \(\dfrac{1}{6}\)

Level 3

Hard

Combine the inverse formula with a product or another function.

Example

Use the table. \(g(1) = 3\) and \(g'(1) = 5\). Let \(h(x) = f^{-1}\!\bigl(g(x)\bigr)\). Find \(h'(1)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

\(h'(x) = \bigl(f^{-1}\bigr)'\!\bigl(g(x)\bigr)\cdot g'(x)\)

\(g(1) = 3\)

\(\bigl(f^{-1}\bigr)'(3) = \dfrac{1}{f'(1)} = \dfrac{1}{4}\)

\(g'(1) = 5\)

\(\dfrac{1}{4}\cdot 5\)

\(\dfrac{5}{4}\)

Answer: \(\dfrac{5}{4}\)

Use the table. Let \(p(x) = x\, f^{-1}(x)\). Find \(p'(3)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

Show Answer

\(p'(x) = f^{-1}(x) + x\,\bigl(f^{-1}\bigr)'(x)\)

\(f^{-1}(3) = 1\)

\(\bigl(f^{-1}\bigr)'(3) = \dfrac{1}{4}\)

Substitute \(x = 3\).

\(1 + 3\cdot\dfrac{1}{4}\)

\(\dfrac{7}{4}\)

Answer: \(\dfrac{7}{4}\)

Let \(f(x) = \ln x\). Find \(\bigl(f^{-1}\bigr)'(2)\).

Show Answer

\(\ln\!\bigl(e^2\bigr) = 2\)

\(f'(x) = \dfrac{1}{x}\)

\(f'\!\bigl(e^2\bigr) = e^{-2}\)

\(\dfrac{1}{e^{-2}}\)

\(e^2\)

Answer: \(e^2\)

Use the table. Find \(\bigl(f^{-1}\bigr)'\!\bigl(f(2)\bigr)\).

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

Show Answer

\(\bigl(f^{-1}\bigr)'\!\bigl(f(a)\bigr) = \dfrac{1}{f'(a)}\)

\(a = 2\)

\(f'(2) = -2\)

\(-\dfrac{1}{2}\)

Answer: \(-\dfrac{1}{2}\)

f is differentiable and one-to-one, \(f(4) = 7\), and \(f'(4) = -5\). Find \(\bigl(f^{-1}\bigr)'(7)\).

Show Answer

\(\bigl(f^{-1}\bigr)'(a) = \dfrac{1}{f'(b)}\ \text{where}\ f(b) = a\)

\(b = 4\)

\(a = 7\)

\(\dfrac{1}{-5}\)

\(-\dfrac{1}{5}\)

Answer: \(-\dfrac{1}{5}\)

Use the table. Which value equals \(\bigl(f^{-1}\bigr)'(5)\)?

\(x\)	\(f(x)\)	\(f'(x)\)
1	3	4
2	5	\(-2\)
3	8	2
4	2	3
5	1	6

A. \(-\dfrac{1}{2}\)

B. \(-2\)

C. \(\dfrac{1}{2}\)

D. \(6\)

Show Answer

\(f(2) = 5\)

\(f'(2) = -2\)

The inverse derivative is the reciprocal.

\(-\dfrac{1}{2}\)

Answer: A. \(-\dfrac{1}{2}\)

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