Question

use logarithmic differentiation to find the derivative of the function.y = x^(9x)

Answer 1

you ahve a choice but simplest is to write
\[ln(y)=9xln(x)\]

Answer 2

then take the derivative using the product rule. you get
\[\frac{d\ln(y)}{dx}=9(\ln(x)+x\times \frac{1}{x})\]
\[\frac{d\ln(y)}{dx}=9(\ln(x)+1)\]

Answer 3

now multiply by the original function to get
\[\frac{dy}{dx}=9x^{9x}(\ln(x)+1)\]

Answer 4

ok this is the new chapter i just still dont understand...im try to..its just stress ful

Answer 5

works because
\[\frac{d}{dx} \ln(f(x))=\frac{f'(x)}{f(x)}\] and so
\[f'(x)=f(x)\times \frac{d}{dx} \ln(f(x))\]

Answer 6

let me say the steps in english. you are trying to find the derivative of a function where the variable is in the exponent

Answer 7

step 1. take the log and pull the exponent out on the ground floor

Answer 8

ok..i get that

Answer 9

step 2 take the derivative of that thing. whatever you have after taking the log

Answer 10

it will almost always involve the product rule since the exponent comes out as a multiplier

Answer 11

step 3 multiply by the original function/
there is not step 4 you are done!

Answer 12

ok thanks

Answer 13

welcome. if you have those 3 steps down it is just the second step that takes any mechanics. finding the derivative of the log. good luck