solve 5. Suppose there exits positive differentiable a(x) and B(x) such thata (x) – B° (x) = 3, anda'(x) = 58(x)(a) Assuming that a-1(x) exists (i.e., the inverse function of a(x) exists), compute an explicitformula for [a(x)]'. Hint: Follow the five step process we used to compute the derivativeof In(x), arctan(x), etc.(b) (Harder) Assuming that B-'(x) exists (i.e., the inverse function of 3(x) exists), computean explicit formula for [3-1(x)]'. Hint: Same basic idea as part (a), but you may need todifferentiate the first property to generate a way to convert B's into as.

Answered: 5. Suppose there exits positive…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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5. Suppose there exits positive differentiable a(x) and B(x) such that
a (x) – B° (x) = 3, and
a'(x) = 58(x)
(a) Assuming that a-1(x) exists (i.e., the inverse function of a(x) exists), compute an explicit
formula for [a(x)]'. Hint: Follow the five step process we used to compute the derivative
of In(x), arctan(x), etc.
(b) (Harder) Assuming that B-'(x) exists (i.e., the inverse function of 3(x) exists), compute
an explicit formula for [3-1(x)]'. Hint: Same basic idea as part (a), but you may need to
differentiate the first property to generate a way to convert B's into as.

Expert Solution

Step 1: Part a)

$G i v e n : α^{3} (x) - β^{3} (x) = 3 (1) α^{'} (x) = 5 β (x) (2) - - - - - - - - - - - - - - - - - - - - - - - - - - T o d e r i v e e x p l i c i t f o r m u l a f o r [α^{- 1} (x)]' .$

$L e t s a y, α^{- 1} (x) = t (3) W e n e e d t o f i n d e x p r e s s i o n f o r \frac{d t}{d x} . - - - - - - - - - - - - - - - - - - - - - - - - - F r o m (3) \begin{array}{rcl} x & = & α (t) \\ D i f f e r e n t i a t e & w . r . t . & x \\ 1 & = & α' (t) . \frac{d t}{d x} \\ \frac{d t}{d x} & = & \frac{1}{α' (t)} \\ R e p l a c e & ' t' & u \sin g (3) \\ \frac{d}{d x} (α^{- 1} (x)) & = & \frac{1}{α' (α^{- 1} (x))} \end{array} H e n c e \Rightarrow \begin{array}{rcl} [α^{- 1} (x)]' & = & \frac{1}{α' (α^{- 1} (x))} \end{array}$