From a Function Value to an Inverse Derivative
When evaluating a derivative of an inverse function, the most important first step is often not differentiating. It is identifying which input of the original function produces the specified input of its inverse.
If \(f(a)=b\), then \(f^{-1}(b)=a\). The inverse derivative formula says that, when the required differentiability and nonzero-derivative conditions hold, the slope of the inverse at \(b\) is the reciprocal of the slope of the original function at \(a\). The derivative is evaluated at \(a\), not at \(b\).
This tutorial focuses on applying that relationship when a function value such as \(f(a)=b\) is known. It does not require finding a formula for the entire inverse function.
How to Locate the Evaluation Point
Suppose a problem asks for \((f^{-1})'(7)\), and you are told \(f(2)=7\). The number \(7\) is the input at which the inverse derivative is requested. Since \(f(2)=7\), the inverse sends \(7\) back to \(2\): \(f^{-1}(7)=2\). Therefore, the original derivative needed in the reciprocal formula is \(f'(2)\).
The calculation follows a consistent sequence:
Use the given information \(f(a)=b\) to identify the original input \(a\) corresponding to the requested inverse input \(b\).
Find \(f'(x)\), then evaluate it at the matched input \(a\).
Provided \(f'(a)\ne 0\) and the inverse is defined locally, calculate \(1/f'(a)\).
Confirm that the original function value matches the requested inverse input and that the reciprocal uses the derivative at the original input.
Why the Reciprocal Rule Works
Let \(g=f^{-1}\) be an inverse defined near the value \(b=f(a)\). The defining relationship between a function and its inverse is \(f(g(y))=y\) for inputs \(y\) near \(b\). Differentiate both sides with respect to \(y\). By the chain rule, the derivative of the left side is \(f'(g(y))g'(y)\), while the derivative of the right side is \(1\).
At \(y=b\), the equality \(g(b)=a\) follows from \(f(a)=b\). Substituting \(y=b\) gives \(f'(a)g'(b)=1\). If \(f'(a)\ne0\), division by \(f'(a)\) is valid, so \(g'(b)=1/f'(a)\), which is the stated formula.
Worked Examples
Worked Example: A Cubic Function with a Known Value
Let \(f(x)=x^3+2x-1\). Find \((f^{-1})'(2)\), given \(f(1)=2\).
Step 1: Match the inputs. The given value \(f(1)=2\) means \(f^{-1}(2)=1\). Thus the original derivative must be evaluated at \(x=1\).
Step 2: Differentiate and evaluate.
Step 3: Take the reciprocal. Since \(f'(1)=5\ne0\), the inverse derivative formula applies:
Check: The function value is \(f(1)=1^3+2(1)-1=1+2-1=2\), so \(1\) is indeed the matching original input. The product of the corresponding slopes is \(f'(1)(f^{-1})'(2)=5\cdot\frac15=1\), as required by the inverse relationship.
Worked Example: A Quadratic on a Restricted Domain
Define \(f(x)=x^2+1\) for \(x>0\). Given \(f(2)=5\), find \((f^{-1})'(5)\).
Step 1: Identify the corresponding input. Because \(f(2)=5\), the inverse sends \(5\) to \(2\): \(f^{-1}(5)=2\). Restricting the original domain to \(x>0\) makes this quadratic one-to-one there, so its inverse is defined on its range.
Step 2: Find the derivative at the original input.
Step 3: Apply the reciprocal rule. Since \(4\ne0\),
Check: Substitution gives \(f(2)=2^2+1=4+1=5\). The original slope times the inverse slope is \(4\cdot\frac14=1\), confirming the reciprocal relation at the corresponding points.
Worked Example: A Function Value and a Linear Derivative
Suppose \(f(3)=10\) and \(f'(x)=2x-1\). Find \((f^{-1})'(10)\), assuming \(f\) has a local inverse near \(x=3\).
Step 1: Match the inverse input. The value \(f(3)=10\) gives \(f^{-1}(10)=3\). Thus we need the original derivative at \(3\), not at \(10\).
Step 2: Evaluate the derivative.
Step 3: Take the reciprocal. Since \(f'(3)=5\ne0\),
Check: The supplied function value pairs input \(3\) with output \(10\). The slope check is \(f'(3)(f^{-1})'(10)=5\cdot\frac15=1\). Both the evaluation point and the reciprocal are consistent with the formula.
A Common Mistake: Using the Inverse Input as the Original Input
The most frequent error is to substitute the requested inverse input directly into \(f'(x)\). That confuses the output value \(b\) with the original input \(a\). The inverse derivative is requested at \(b\), but the denominator is \(f'(a)\), where \(f(a)=b\).
For the restricted quadratic \(f(x)=x^2+1\), with \(f(2)=5\), the incorrect calculation would be:
This uses \(5\) as though it were the original input. But \(5\) is the original function's output, and \(f(2)=5\), so the matching input is \(2\). The correction is:
Check the corrected value: \(f(2)=5\), and \(f'(2)\cdot\frac14=4\cdot\frac14=1\). The incorrect value would give \(f'(2)\cdot\frac1{10}=4\cdot\frac1{10}=\frac25\), not \(1\).
When This Method Applies
The method is efficient when the problem supplies a function value and enough information to evaluate the original derivative at the matching input. The exact formula for \(f^{-1}\) is unnecessary. If \(f(a)=b\) and \(f'(a)\) is known or can be computed, the inverse derivative at \(b\) follows directly, as long as the local inverse and nonzero-slope conditions are satisfied.
Before giving an answer, check these points:
- Does the given function value really state \(f(a)=b\), with \(b\) equal to the requested inverse input?
- Did you evaluate \(f'\) at the original input \(a\), rather than at \(b\)?
- Is \(f'(a)\ne0\), so taking its reciprocal is valid?
- Is an inverse defined near the point in question, at least locally?
- Does the product \(f'(a)(f^{-1})'(b)\) equal \(1\)?
These checks are particularly useful when the numbers in the question make it tempting to substitute the same value into both the original and inverse derivatives. Remember that an inverse swaps input and output, so the two evaluation points are generally different.
Check Your Understanding
For each question, identify the original input that corresponds to the inverse input before applying the reciprocal formula.
- If \(f(4)=9\) and \(f'(4)=6\), find \((f^{-1})'(9)\). State the nonzero-derivative check.
- If \(g(2)=-3\) and \(g'(2)=-5\), find \((g^{-1})'(-3)\). Verify the product of the slopes.
- Suppose \(h(1)=6\) and \(h'(x)=x^2+2\). Find \((h^{-1})'(6)\), showing the derivative evaluation.
- A student is given \(p(3)=8\) and \(p'(3)=4\), and claims \((p^{-1})'(8)=1/p'(8)\). Explain the error and give the correct expression and value.
- What condition on \(f'(a)\) is required to use \((f^{-1})'(b)=1/f'(a)\) when \(f(a)=b\), and why?
