How to differentiate y= (ln(4x^2))/(3^(6x) ?

1 Answer
May 29, 2018

(2-6x ln 3 ln(4x^2))/(3^(6x)x)

Explanation:

You can use the quotient rule

d/dx (u/v) = (v (du)/dx - u(dv)/dx)/v^2

Here

u = ln(4x^2) = ln 4+2ln x implies
(du)/dx = 2/x

and

v = 3^(6x) = (e^{ln 3})^{6x} = e^{6 (ln 3 )x}implies

(dv)/dx = 6 ln 3 times e^{6 (ln 3 )x} = 6 ln 3 times 3^(6x)

Hence

d/dx(ln(4x^2)/3^(6x)) =(3^(6x)times 2/x-ln(4x^2)times 6 ln 3 times 3^(6x))/(3^(6x))^2
qquad = (2-6x ln 3 ln(4x^2))/(3^(6x)x)