how would I find the derivative of a natural log funciton?

Question

h(w)= w^6 *ln(16w)

 

I attempted this by using the product and the ln rule.

I got h'(w)= 6w(ln(16w))*w^6 *1/16w *16

I then reduced the 16's and the w and I got

h'(w)= 6(ln(16w))*w^6

but I got it wrong. could you please explain where I went wrong. I just learned the ln rule but I think I used it correctly in this instance.

Answer 1

h(w)= w⁶ln(16w)

 

Use the product rule.  

 

h = uv, h' = u'v + uv'

 

u = w⁶, v = ln(16w)

 

For the second term of the product rule, use the ln rule and the chain rule:

 

v' = [1/(16w)] * 16

 

dh/dw = 6w⁵ln(16w) + w⁶[1/(16w)](16) 

 

You can simplify it from here.

Answer 2

h'(w) = 6w⁵ln(16w) + w⁶(16 / (16w))

 

h'(w) = 6w⁵ln(16w) + w⁵

 

h'(w) = w⁵(6ln(16w) + 1)