Question

f(x)=x^(3x)
Use logarithmic differentiation to determine the derivative.

f'(x)=
f'(1)=

Answer 1

Change the f(x) to y for convenience, take ln of both sides:
\[\Large \ln y= \ln x^{3x}\] then bring down the 3x \[\Large \ln y= 3x \ln x\] Now use implicit differentiation - derivative of ln y, but don't forget the derivative of y is dy/dx

then use the product rule like last time on the right\\[\Large \left( \frac{ 1 }{ y } \right)\frac{ dy }{ dx } = 3 \ln x + 3x \left( \frac{ 1 }{ x } \right)\]

Answer 2

so then i solve for dy/dx

Answer 3

Yes, and remember that the y in that equation can be replaced by x^(3x)

Answer 4

alright give me a minute to try it

Answer 5

just to make sure i can rewite it as this right [3lnx+3x/x]y which it would end up as [3lnx+3x/x](x^3x)

Answer 6

Yeah and you can simplify the 3x/x
\[\LARGE \frac{ dy }{ dx } = x^{3x}\left( 3 \ln x + 3 \right)\]

Answer 7

oh ok then from there i just plug in 1 right

Answer 8

Yep :)