Use the results of Exercise 81 to show that the function have inverses over their domains. Find
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Use the results of Exercise 81 to show that the function have inverses over their domains. Find a formula for dƒ -1/dx using Theorem 3
ƒ(x) = 27x3
Exercise 81
Show that increasing functions and decreasing functions are one to- one. That is, show that for any x1 and x2 in I, x2 ≠ x1 implies ƒ(x2) ≠ ƒ(x1).
Theorem 3
Transcribed Image Text:
THEOREM 3-The Derivative Rule for Inverses If f has an interval I as do- main and f'(x) exists and is never zero on I, then f-¹ is differentiable at every point in its domain (the range of f). The value of (f¹)' at a point b in the domain of f¹ is the reciprocal of the value of f' at the point a = f-¹(b): (f-¹)'(b) or df-1 dx x=b = = 1 ƒ'(ƒ-¹(b)) df dx x=f-¹(b) (1)
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Related Book For
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
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