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) = x^5/3

Exercise 81

Show that increasing functions and decreasing functions are oneto- 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)