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Understanding Derivatives of Inverse Functions: The Reciprocal Relationship

Explore the relationship between the derivatives of a function and its inverse, particularly focusing on linear functions. Discover how the slopes of f and f-1 are reciprocals of each other and what this implies. See how this reciprocal relationship holds when we zoom in on corresponding points. In general, when f and f-1 are inverse functions, their derivatives at corresponding points are reciprocals of one another.

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Understanding Derivatives of Inverse Functions: The Reciprocal Relationship

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  1. The derivatives of f and f -1 How are they related?

  2. (4,6) Recall that if we have a one-to-one function f, we get f -1 from f, we switch every x and y coordinate. f -1 (2,3) (6,4) f (3,2) (-1,-1)

  3. Inverses of Linear functions In other words, the inverse of a linear function is a linear function and the slope of the function and its inverse are reciprocals of one another.

  4. Inverses of Linear functions In other words, the inverse of a linear function is a linear function and the slope of the function and its inverse are reciprocals of one another.

  5. f -1 Slope is f Slope is m.

  6. What about the more general question? What is the relationship between the slope of f and the slope of f -1? f -1 (2,3) f (3,2)

  7. Note: the points where we should be comparing slopes are “corresponding” points. E.g. (3,2) and (2,3). f -1 (2,3) f (3,2)

  8. What happens when we “zoom in” on these points? f -1 (2,3) f (3,2) We see straight lines whose slopes are reciprocals of one another!

  9. f -1 In general, what does this tell us about the relationship between and ? (b, f -1(b)) f (a, f (a)) But a = f -1(b), so . . .

  10. Upshot If f and f-1 are inverse functions, then their derivatives at “corresponding” points are reciprocals of one another :

  11. Derivative of the logarithm f (x) = e x (a, e a) f -1(x) = ln(x) (b, ln(b))

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