when taking the derivative of the natural log you do 1/x *x' correct?

Question

I have these natural log questions and some of them I get correct and some of them I don't. I am using the same rule for each so I don't know why I am getting things wrong.

for example: I was given w⁶ln12w

I took the derivative and got 6w⁵ln(12w) + w⁵

I used all the same rules as previous ones I got right, so I don't know where I am going wrong

 

also there is another part that is messing with me. I was given h(z)=z^ln4+ln3

ln3 is just a constant so it cancels and I preceded to take the natural log to get

ln(z) (z^ln4)

what am I doing wrong?

Answer 1

f(w) = w⁶ln(12w)

 

Use product rule, with chain rule for the second term.

 

f'(w) = 6w⁵ · ln(12w) + w⁶[1/(12w)] · 12 = 6w⁵ · ln(12w) + w⁵

 

I get the same answer as you.

 

h(z)=z^ln4 + ln3

 

Use the power rule here, since the exponent is a constant.

 

h'(z) = (ln4)z^{ln4 - 1}

 

 

Answer 2

The derivative f'(x) of f(x) = ln |x| = 1/x* dx

 

In your first example, you need to use the Product Rule, which says that given f(x) * g(x), the derivative of this product is:

 

h(x) = f'(x) * g(x) + g'(x) * f(x)

 

The derivative of ln (12w) = 1/12w * 12 = 1/w