The derivatives of f and f -1

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The derivatives of f and f -1

• How are they related?

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(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)
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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.
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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.
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f -1
f
Slope is m.
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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)
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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)
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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!
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f -1
(b, f -1(b))
f
(a, f (a))
But a f -1(b), so . . .
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Upshot
If f and f -1 are inverse functions, then their
derivatives at corresponding points are
reciprocals of one another
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Derivative of the logarithm
f (x) e x
(a, e a)
f -1(x) ln(x)
(b, ln(b))