Question

How do you use the chain rule to differentiate `ln(4x)^10`?

Answer 1

Please see below.

Explanation:

My preference would not be the chain rule, but here's how to differentiate using the chain rule.

Use `d/dx(lnu) = 1/u*(du)/dx`.

`d/dx(ln(4x)^10) = d/dx(ln(underbrace((4x)^10)_u))`

`= 1/(underbrace((4x)^10)_u) * d/dx(underbrace((4x)^10)_u)`

`= 1/(4x)^10 * 10(4x)^9 * d/dx(4x)`

`= 1/(4x)^10 * 10(4x)^9 * 4`

`= (40(4x)^9)/(4x)^10`

`= 4/(4x) = 1/x`.

Using properties of logarithms to rewrite first

`ln(4x)^10 = 10 ln(4x) = 10(ln4+lnx) = 10ln4+10lnx`.

Note that `10ln4` is a constant, so its derivative is `0`.

`d/dx(ln(4x)^10) = d/dx(10ln4+10lnx)`

`= 0+10*1/x`

`= 10/x`.