How do you differentiate $log_3(9x^2sin(9x^2) )$ ?

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Answer 1 · 2017-03-31T21:02:05

see below

Use the following Properties of Logarithm to expand the problem before taking derivatives.

Then use the formula $color(red)(d/dx(log_bx)=1/(xln b)$ to find the derivative

$y=log_3(9x^2sin(9x^2))$

$y=log_3 9 + log_3x^2+log_3 sin(9x^2)$

$y=log_3 9 + 2 log_3x+log_3 sin(9x^2)$

$color(blue)(y'=0+2/(xln3)+1/(sin(9x^2)ln3) * cos (9x^2)*18x$

$color(blue)(y'=2/(xln3)+(cos (9x^2)*18x)/(sin(9x^2)ln3)$

$color(blue)(y'=2/(xln3)+((18x) cot(9x^2))/ln3$

$color(blue)(y'=((18x^2) cot(9x^2))/(xln3)$

How do you differentiate log_3(9x^2sin(9x^2) ) log_3(9x^2sin(9x^2) ) ?