Calculus I · 2A · lesson
Derivative of an Inverse Function
Learn derivative of an inverse function with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Implicit, inverse, and logarithmic differentiationWhat this section is building
Learn derivative of an inverse function with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Implicit equations constrain variables together; inverse functions exchange inputs and outputs; logarithms turn products and powers into sums.
Choose implicit, inverse, or logarithmic differentiation from the equation's representation, not from surface complexity.
Dropping a y-prime factor, using a reciprocal slope at the wrong point, or ignoring domain restrictions.
Learning objectives
Use the inverse-function derivative theorem numerically and symbolically.
Reciprocal Slopes of Inverse Functions
Before the formulas
In Derivative of an Inverse Function, inverse and implicit ideas meet. Swapping input and output swaps horizontal and vertical change, so inverse slopes are reciprocals at corresponding points. Taking logarithms can also reveal hidden structure by turning products into sums and exponents into coefficients.
These methods are strategic transformations, not new definitions of derivative. State the domain assumptions, preserve the original relationship, and substitute back at the end. A clean solution explains why the transformation helps before carrying out the algebra.
Read this graph as text
Inverse-function slopes are reciprocal because coordinates swap. Reflecting a graph across y=x swaps horizontal and vertical changes. A tangent with slope m reflects to one with slope 1/m . The point (2,4) on y=x 2 reflects to (4,2) on y= x . The tangent slope 4 reflects to slope 1/4 . This geometric swap explains the inverse derivative formula: the inverse rate is the reciprocal of the original rate, evaluated at the matching input.
The visual uses labeled positions, solid and dashed line styles, and written descriptions so inverse-function slopes are reciprocal because coordinates swap does not depend on color.
Why it matters: This visual provides the geometric justification for (f -1 )'(a)=1/f'(f -1 (a)) . It must make clear that the two derivatives are evaluated at corresponding points, not at the same horizontal coordinate.
Reflecting a graph across y=x swaps horizontal and vertical changes. A tangent with slope m reflects to one with slope 1/m.
Inverse functions swap inputs and outputs, so slopes become reciprocals
The graph of an inverse function is the reflection of the original graph across . A tangent line with rise for run becomes a reflected line with rise for run , giving reciprocal slope .
The derivative must be evaluated at corresponding points. If , then the inverse derivative at input depends on , not on . Keeping the point pair visible prevents the usual substitution mistake.
Inverse functions exchange inputs and outputs, so their graphs reflect across the line . The slope at a reflected point becomes a reciprocal: a steep original graph produces a shallow inverse graph, provided the original slope is nonzero.
The rule is local. A function must actually be one-to-one on a suitable interval, and the derivative at the corresponding original point cannot vanish. Those conditions are what allow the inverse to behave like a smooth function nearby.
If is one-to-one and differentiable with , then
Inverse graphs reflect across , which swaps rise and run. Their tangent slopes are reciprocal at corresponding points.
Use inverse data without finding a formula
Suppose and . Find .
Worked solution
Write a real attempt before opening the supplied answer.
The identity differentiates to
Solving gives the inverse derivative formula.
inverse-function-01If and , find .
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Show hint
The inverse slope is the reciprocal of the original slope at the corresponding point.
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Inverting a calibration curve
A laboratory instrument maps concentration to signal . Suppose and signal units per mg/L. The inverse calibration converts signal back to concentration, and
A one-unit signal error near therefore produces about mg/L concentration error.
Local invertibility and nonzero slope
The condition does more than prevent division by zero in the inverse derivative formula. Under suitable continuity assumptions, it means the function is locally one-to-one near , so a differentiable inverse exists nearby. This one-dimensional fact becomes the Inverse Function Theorem for maps between higher-dimensional spaces.
inverse-extra-01If and , find .
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Show hint
Use the reciprocal slope at the corresponding input.
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Submit an answer first. The hint is available now.
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