First of all, separate the natural logarithms, using the rule
Apply the rule
Now differentiate each natural logarithm separately using the chain rule.
Now subtract them to find the derivative of the entire function.
Hopefully this helps!
We are dealing with two function compositions and a fraction, so we will require the use of the chain and quotient rules. The chain rule states that if
The quotient rule states that if
We know that
Then, to calculate the derivative of the second term, use the quotient rule:
Finally, we will have to find the derivative of
Using the chain rule, with
Substituting in the middle expression gives:
This looks quite messy. We can simplify by multiplying and dividing the second term in the numerator, by
fraction into a complex fraction of the form
Much better now. Finally, we can substitute this into the initial expression:
We could even factor the quadratic on the bottom to get: