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Math Coach Amy
Amy Ferguson Moncure

Calculus BC · Unit 3

Find g'(a) for an Inverse

Use a few values of f, or a formula for f, to find the derivative of the inverse at one point.

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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

The inverse derivative is a reciprocal

If g(x) = f⁻¹(x) and f(b) = a,

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

Use f′ at b. That b is the input that produces a.

Match the output before you differentiate

Several values may be listed. Keep the input whose output is a. A different input is not b.

A stated value of g names b

If you are told g(a) = b, then b is the input.

Check by computing f(b). It should equal a.

Then take the reciprocal

Differentiate f, substitute b, and flip that number.

A negative derivative stays negative after the flip.

Level 1

Easy

One input, its output, and the derivative at that input.

Example
Let \(g(x) = f^{-1}(x)\). \(f(2) = 5\) and \(f'(2) = 4\). Find \(g'(5)\).

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

\(f(2) = 5\), so \(b = 2\).

\(f'(2) = 4\)

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

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

1
Let \(g(x) = f^{-1}(x)\). \(f(3) = 7\) and \(f'(3) = 2\). Find \(g'(7)\).
Show Answer

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

\(f(3) = 7\), so \(b = 3\).

\(f'(3) = 2\)

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

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

2
Let \(g(x) = f^{-1}(x)\). \(f(1) = 4\) and \(f'(1) = 5\). Find \(g'(4)\).
Show Answer

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

\(f(1) = 4\), so \(b = 1\).

\(f'(1) = 5\)

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

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

3
Let \(g(x) = f^{-1}(x)\). \(f(-2) = 6\) and \(f'(-2) = 3\). Find \(g'(6)\).
Show Answer

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

\(f(-2) = 6\), so \(b = -2\).

\(f'(-2) = 3\)

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

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

4
Let \(g(x) = f^{-1}(x)\). \(f(4) = -1\) and \(f'(4) = 8\). Find \(g'(-1)\).
Show Answer

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

\(f(4) = -1\), so \(b = 4\).

\(f'(4) = 8\)

\(\dfrac{1}{8}\)

Answer: \(\dfrac{1}{8}\)

5
Let \(g(x) = f^{-1}(x)\). \(f(0) = 9\) and \(f'(0) = 6\). Find \(g'(9)\).
Show Answer

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

\(f(0) = 9\), so \(b = 0\).

\(f'(0) = 6\)

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

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

Level 2

Medium

Two inputs are given. Use the input whose output equals a.

Example
Let \(g(x) = f^{-1}(x)\). \(f(5) = 2\), \(f(1) = 8\), \(f'(5) = 4\), and \(f'(1) = 3\). Find \(g'(2)\).

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

\(f(5) = 2\), so \(b = 5\).

\(f(1) = 8\) is not \(2\).

\(f'(5) = 4\)

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

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

1
Let \(g(x) = f^{-1}(x)\). \(f(6) = -1\), \(f(4) = 3\), \(f'(6) = 2\), and \(f'(4) = 5\). Find \(g'(-1)\).
Show Answer

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

\(f(6) = -1\), so \(b = 6\).

\(f(4) = 3\) is not \(-1\).

\(f'(6) = 2\)

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

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

2
Let \(g(x) = f^{-1}(x)\). \(f(-3) = 4\), \(f(2) = 7\), \(f'(-3) = -2\), and \(f'(2) = 6\). Find \(g'(4)\).
Show Answer

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

\(f(-3) = 4\), so \(b = -3\).

\(f(2) = 7\) is not \(4\).

\(f'(-3) = -2\)

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

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

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

3
Let \(g(x) = f^{-1}(x)\). \(f(0) = -5\), \(f(7) = 2\), \(f'(0) = 4\), and \(f'(7) = -8\). Find \(g'(2)\).
Show Answer

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

\(f(7) = 2\), so \(b = 7\).

\(f(0) = -5\) is not \(2\).

\(f'(7) = -8\)

\(\dfrac{1}{-8}\)

\(-\dfrac{1}{8}\)

Answer: \(-\dfrac{1}{8}\)

4
Let \(g(x) = f^{-1}(x)\). \(f(8) = 3\), \(f(-4) = -6\), \(f'(8) = 1\), and \(f'(-4) = 2\). Find \(g'(-6)\).
Show Answer

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

\(f(-4) = -6\), so \(b = -4\).

\(f(8) = 3\) is not \(-6\).

\(f'(-4) = 2\)

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

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

5
Let \(g(x) = f^{-1}(x)\). \(f(2) = 9\), \(f(9) = 2\), \(f'(2) = -3\), and \(f'(9) = 6\). Find \(g'(9)\).
Show Answer

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

\(f(2) = 9\), so \(b = 2\).

\(f(9) = 2\) is not \(9\).

\(f'(2) = -3\)

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

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

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

Level 3

Hard

f is a formula, and you are told g(a). That number is b.

Example
Let \(f(x) = x^3 + 2x + 1\) and \(g(x) = f^{-1}(x)\). \(g(4) = 1\). Find \(g'(4)\).

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

\(g(4) = 1\), so \(b = 1\).

Check.

\(f(1) = 1^{3} + 2(1) + 1\)

\(4\)

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

\(f'(1) = 3(1)^{2} + 2\)

\(5\)

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

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

1
Let \(f(x) = x^3 + 4x\) and \(g(x) = f^{-1}(x)\). \(g(5) = 1\). Find \(g'(5)\).
Show Answer

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

\(g(5) = 1\), so \(b = 1\).

\(f(1) = 1^{3} + 4(1)\)

\(5\)

\(f'(x) = 3x^{2} + 4\)

\(f'(1) = 3(1)^{2} + 4\)

\(7\)

\(\dfrac{1}{7}\)

Answer: \(\dfrac{1}{7}\)

2
Let \(f(x) = x^3 + 6x - 2\) and \(g(x) = f^{-1}(x)\). \(g(-9) = -1\). Find \(g'(-9)\).
Show Answer

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

\(g(-9) = -1\), so \(b = -1\).

\(f(-1) = (-1)^{3} + 6(-1) - 2\)

\(-1 - 6 - 2\)

\(-9\)

\(f'(x) = 3x^{2} + 6\)

\(f'(-1) = 3 + 6\)

\(9\)

\(\dfrac{1}{9}\)

Answer: \(\dfrac{1}{9}\)

3
Let \(f(x) = x^5 + x\) and \(g(x) = f^{-1}(x)\). \(g(2) = 1\). Find \(g'(2)\).
Show Answer

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

\(g(2) = 1\), so \(b = 1\).

\(f(1) = 1^{5} + 1\)

\(2\)

\(f'(x) = 5x^{4} + 1\)

\(f'(1) = 5(1)^{4} + 1\)

\(6\)

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

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

4
Let \(f(x) = x^3 + x\) and \(g(x) = f^{-1}(x)\). \(g(10) = 2\). Find \(g'(10)\).
Show Answer

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

\(g(10) = 2\), so \(b = 2\).

\(f(2) = 8 + 2\)

\(10\)

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

\(f'(2) = 12 + 1\)

\(13\)

\(\dfrac{1}{13}\)

Answer: \(\dfrac{1}{13}\)

5
Let \(f(x) = 2x^3 + x\) and \(g(x) = f^{-1}(x)\). \(g(18) = 2\). Find \(g'(18)\).
Show Answer

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

\(g(18) = 2\), so \(b = 2\).

\(f(2) = 2(8) + 2\)

\(18\)

\(f'(x) = 6x^{2} + 1\)

\(f'(2) = 24 + 1\)

\(25\)

\(\dfrac{1}{25}\)

Answer: \(\dfrac{1}{25}\)

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