Question

How do I find the derivative of `f(x)=(16x^2+1) ln (x-3)`?

Answer 1

This question will demand the use of chain rule and product rule, as well as the knowledge of how to derivate a logarithmic function.

Explanation:

First of all, we have a product of two terms:

• `16x^2+1`
• `ln(x-3)`

However, one cannot derivate straightforward a logarithmic function with a function in it. Instead, we must apply chain rule and rename `u=x-3` so now we can proceed. The chain rule states that `(dy)/(dx)=(dy)/(du)(du)/(dx)`, so:

`(dy)/(dx)=(1/u)(1)=1/u=1/(x-3)`

Fine. Now, we can apply the product rule, which states that, for `f(x)=g(x)*h(x)`, `f'(x)=g'(x)*h(x)+g(x)*h'(x)`, as follows:

(where the symbol `'` indicates the first derivative)

`f'(x)=32xln(x-3)+(16x^2+1)(1/(x-3))`

`f'(x)=32xln(x-3)+(16x^2+1)/(x-3)`

The struggle to develop and try to simplify it is not worth it. This is your final answer!