Question

Derivative Calculus problem that I have done over ten times and can’t seem to get

Answer 1

You have the following expression:

`(-4x^2+16)/((x^2+4)^2)`

The previous expression can be written as follow:

`(-4x^2+16)\cdot(1)/((x^2+4)^2)`

The derivative of the previous expression is the derivative of a product:

`\begin{gathered} (-4x^2+16)^(\prime)\cdot(1)/((x^2+16)^2)+(-4x^2+16)\cdot((1)/((x^2+4)^2))^(\prime) \\ =(-8x)\cdot(1)/((x^2+16)^2)+(-4x^2+16)\cdot(-2)(x^2+4)^(-3)\cdot(2x) \\ =(-8x)/((x^2+16)^2)-(4x(-4x^2+16))/((x^2+4)^3) \end{gathered}`

by factorizing the numerator of the second term, you obtain:

`\begin{gathered} (-8x)/((x^2+16)^2)+(4x(x^2-16))/((x^2+4)^3) \\ =(-8x)/((x^2+16)^2)+(4x(x+4)(x-4))/((x^2+4)^3) \\ =(-8x)/((x^2+16)^2)+(4x(x-4))/((x^2+4)^2) \end{gathered}`