Reading an Inverse Derivative from a Table
A table may give values of a function \(f\) and its derivative \(f'\) at several inputs. To find the derivative of the inverse at a specified number, you need to read the table in the right direction: first find the input whose function value equals that number, then use the derivative at that input.
The distinction matters because the input to \(f^{-1}\) is an output of \(f\). If \(f(3)=8\), then \(f^{-1}(8)=3\). The inverse derivative at \(8\) depends on \(f'(3)\), not on a derivative value at input \(8\).
What the Table Entries Mean
A typical table has one column for inputs \(x\), one for function values \(f(x)\), and one for derivative values \(f'(x)\). A row records three facts about the same input. For example, if a row says \(x=2\), \(f(x)=5\), and \(f'(x)=3\), then \(f(2)=5\) and \(f'(2)=3\).
Suppose the question asks for \((f^{-1})'(5)\). Since \(5\) is the input to the inverse, look in the \(f(x)\) column for the value \(5\). The matching row tells you \(x=2\), and that row’s derivative entry is \(f'(2)=3\). Therefore, the inverse derivative is \(1/3\).
A reliable approach keeps the two jobs separate: use the function-value column to find the input-output pair, and use the derivative column to find the slope at the original input. Do not search for the requested number in the \(x\) column unless the question specifically asks about that input to \(f\).
Worked Example: Finding the Matching Row
Worked Example: Finding the Matching Row
The table gives values of \(f\) and \(f'\). Find \((f^{-1})'(5)\).
| \(x\) | \(f(x)\) | \(f'(x)\) |
|---|---|---|
| \(-1\) | 3 | 2 |
| 0 | 1 | \(-4\) |
| 2 | 5 | 3 |
| 4 | 9 | \(-2\) |
First locate \(5\) in the \(f(x)\) column. It appears in the row with \(x=2\), so \(f(2)=5\). The derivative in that row is \(f'(2)=3\). Since this derivative is nonzero, the inverse derivative rule gives
Check the matching pair: \(f(2)=5\), so \(f^{-1}(5)=2\). Check the reciprocal: \(3\cdot\frac13=1\). Thus the table’s original slope and the inverse slope are reciprocal, as required.
A Brief Justification of the Reciprocal
The reciprocal rule follows from the definition of a derivative. This argument also shows why the nonzero-slope condition matters. Let \(f(x_0)=y_0\), and suppose \(f\) has a local inverse \(g=f^{-1}\) near \(y_0\). Assume \(g\) is continuous at \(y_0\), \(f\) is differentiable at \(x_0\), and \(f'(x_0)\ne0\). For \(y\ne y_0\) near \(y_0\), set \(x=g(y)\). Then \(f(x)=y\), and \(g(y_0)=x_0\).
The difference quotient for the inverse can now be rewritten using these equalities:
As \(y\) approaches \(y_0\), continuity of \(g\) implies \(x=g(y)\) approaches \(x_0\). The denominator in the last expression therefore approaches \(f'(x_0)\). Because \(f'(x_0)\ne0\), its reciprocal approaches \(1/f'(x_0)\). Hence the inverse difference quotient has a limit, and
For typical table questions, the function is understood to have an inverse on the relevant interval. A continuous function that is strictly monotonic on an interval has a continuous inverse on its range. The table supplies the values needed to apply the rule; a finite table by itself does not prove that the function has an inverse or that it is differentiable nearby.
Worked Example: A Negative Derivative
Worked Example: A Negative Derivative
The function \(h\) is invertible on the interval in question. Use the table to find \((h^{-1})'(4)\).
| \(x\) | \(h(x)\) | \(h'(x)\) |
|---|---|---|
| \(-2\) | 6 | 5 |
| 1 | \(-3\) | 2 |
| 3 | 4 | \(-\frac{5}{2}\) |
| 6 | 9 | 1 |
The requested inverse input is \(4\). Find \(4\) in the \(h(x)\) column: it occurs when \(x=3\). Thus \(h(3)=4\), and the derivative at that original input is \(h'(3)=-\frac52\). Apply the reciprocal rule:
The result is negative because the original function’s slope at \(x=3\) is negative. Check the reciprocal by multiplication: \((-\frac52)(-\frac25)=1\). The matching pair also checks: \(h(3)=4\), so \(h^{-1}(4)=3\).
Worked Example: Keeping Inputs and Outputs Straight
Worked Example: Keeping Inputs and Outputs Straight
A table lists a function \(p\) and its derivative. Find \((p^{-1})'(7)\).
| \(x\) | \(p(x)\) | \(p'(x)\) |
|---|---|---|
| 2 | 1 | 4 |
| 4 | 7 | \(-2\) |
| 7 | 10 | 3 |
| 9 | 12 | 5 |
The number \(7\) is the input to \(p^{-1}\), so find \(7\) in the \(p(x)\) column. It occurs at \(x=4\), which means \(p(4)=7\). The same row gives \(p'(4)=-2\). Therefore,
Check the input-output match: \(p(4)=7\), so \(p^{-1}(7)=4\). Check the reciprocal: \((-2)(-\frac12)=1\). The table entry \(p'(7)=3\) is not used: it describes the slope of \(p\) at input \(7\), not at the input \(4\) that maps to \(7\).
When the Table Is Not Enough
A table gives information at selected inputs, but it does not automatically establish every condition needed for the inverse derivative rule. Before reporting a value, check the following:
- The matching function value is present. To find \((f^{-1})'(y_0)\), the table must identify an input \(x_0\) with \(f(x_0)=y_0\).
- The corresponding derivative is nonzero. If \(f'(x_0)=0\), the reciprocal formula cannot be used to obtain a finite inverse derivative there.
- The inverse is meaningful on the relevant interval. The function must be one-to-one there, so that an output determines a single input.
If two different listed inputs have the same function value, the table alone does not identify a unique inverse value unless the problem specifies a restricted interval or branch. Similarly, if the relevant derivative entry is missing, the table does not provide enough information to calculate the inverse derivative from this rule.
Common Mistake: Using the Wrong Row or Slope
Suppose a question asks for \((p^{-1})'(7)\), and the table says \(p(4)=7\) and \(p'(4)=-2\). A common incorrect calculation is
This has found the correct row but has forgotten to take the reciprocal. Another possible error is to use \(p'(7)=3\), which is the slope at input \(7\), even though \(p(7)\) is not the output being inverted. The correct calculation is
A quick check catches the first error: the original derivative \(-2\) and the inverse derivative should multiply to \(1\). But \((-2)(-2)=4\), not \(1\), whereas \((-2)(-\frac12)=1\).
A Compact Table-Reading Routine
In \((f^{-1})'(y_0)\), identify \(y_0\), the input to the inverse function.
Locate \(y_0\) in the \(f(x)\) column and read its corresponding original input \(x_0\).
From that same row, take \(f'(x_0)\), and confirm that it is not zero.
Calculate \(1/f'(x_0)\); verify that multiplying this value by \(f'(x_0)\) gives \(1\).
The essential habit is to follow the row across the table. The function-value entry identifies which original input maps to the requested inverse input; the derivative entry in that same row supplies the slope to reciprocate.
Check Your Understanding
For each question, identify the matching input first, then apply the inverse derivative rule where its conditions are met.
- A table gives \(q(2)=6\) and \(q'(2)=4\). Find \((q^{-1})'(6)\).
- A table gives \(r(5)=1\) and \(r'(5)=-3\). Find \((r^{-1})'(1)\).
- If \(s(3)=8\), which table entry is needed to calculate \((s^{-1})'(8)\): \(s'(3)\) or \(s'(8)\)? Explain.
- A table gives \(t(4)=9\) and \(t'(4)=0\). Can the reciprocal rule give a finite value for \((t^{-1})'(9)\)? Why or why not?
- A table lists \(u(1)=5\) and \(u(3)=5\). What issue must be resolved before using the table to identify \(u^{-1}(5)\)?
