Find ifIn 9x+x6 In x?(m(4))'r =Indrdx

Given that: The equation is r = ln 9xx6 ln x2 + ln 4x3 .

Answered: dr dæ | bartleby

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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Question

Find if
In 9x
+
x6 In x?
(m(4))'
r =
In
dr
dx

Expert Solution

Step 1

Given that:

The equation is $r = \frac{\ln 9 x}{x^{6} \ln x^{2}} + {(\ln (\frac{4}{x}))}^{3}$ .

Step 2

By using,

Quotient rule of derivative :

$\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d}{d x} u - u \frac{d}{d x} v}{v^{2}}$

Chain rule of derivative :

$\frac{d}{d x} f (g (x)) = f^{'} (g (x)) . \frac{d}{d x} g (x)$

Step 3

Differentiate the above equation with respect to x,

$\frac{d r}{d x} = \frac{d}{d x} (\frac{\ln 9 x}{x^{6} \ln x^{2}} + {(\ln (\frac{4}{x}))}^{3}) = \frac{d}{d x} (\frac{\ln 9 x}{x^{6} \ln x^{2}}) + \frac{d}{d x} {(\ln (\frac{4}{x}))}^{3} B y u \sin g q u o t i e n t r u l e o f d e r i v a t i v e, \frac{d}{d x} (\frac{\ln 9 x}{x^{6} \ln x^{2}}) = \frac{x^{6} \ln x^{2} \frac{d}{d x} (\ln 9 x) - (\ln 9 x) \frac{d}{d x} (x^{6} \ln x^{2})}{{(x^{6} \ln x^{2})}^{2}} = \frac{x^{6} \ln x^{2} . \frac{9}{9 x} - (\ln 9 x) (6 x^{5} \ln x^{2} + 2 x^{5})}{{(x^{6} \ln x^{2})}^{2}} = \frac{x^{5} \ln x^{2} - (\ln 9 x) (6 x^{5} \ln x^{2} + 2 x^{5})}{x^{12} {(\ln x^{2})}^{2}}$

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