Question

What is the derivative of `f(x)=(x^2-4)ln(x^3/3-4x)`?

Answer 1

We'll need the product rule for `f(x)` and the chain rule for the second term `ln(x^3/3-4x)`

Explanation:

• Product rule: be `y=f(x)g(x)`, then `(dy)/(dx)=f'(x)g(x)+f(x)g'(x)`
• Chain rule: `(dy)/(dx)=(dy)/(du)(du)/(dx)`

`(df(x))/(dx)=(2x)ln(x^3/3-4x)+(x^2-4)(1/(x^3/3-4x)(x^2-4))`

`(df(x))/(dx)=2xln(x^3/3-4x)+(x^2-4)^2/((x^3-12x)/3)`

`(df(x))/(dx)=2xln(x^3/3-4x)+(3(x^2-4)^2)/(x(x^2-12))`