# What is the derivative of # ln (x^8)/ x^2#?

##### 1 Answer

Dec 17, 2015

You can do this using the product rule or the quotient rule. The product rule can be easier to simplify.

These two derivatives will be used overall:

#d/(dx)[ln(x^8)] = (8x^7)/(x^8) = 8/x#

#d/(dx)[1/x^2] = -2/x^3#

So, what we can do is:

**PRODUCT RULE**

#\mathbf(d/(dx)[g(x)h(x)] = g(x)(dh(x))/(dx) + h(x)(dg(x))/(dx))#

#d/(dx)[ln(x^8)*1/x^2] = ln(x^8)(-2/x^3) + 1/x^2 8/x#

#= -(2ln(x^8))/x^3 + 8/x^3#

#= color(blue)(1/x^3 [8 - 2ln(x^8)])#