Question

I have been struggling with this problem for around 2 hours and can’t seem to get it

Answer 1

the quotient rule say:

`((f(x))/(g(x)))^(\prime)=(g(x)\cdot f^(\prime)(x)-f(x)\cdot g^(\prime)(x))/((g(x))^2)`

now we defined:

`\begin{gathered} f(x)=-4x^2+16 \\ g(x)=(x^2+4)^2 \end{gathered}`

and the derivative:

`\begin{gathered} f^(\prime)(x)=-8x \\ g^(\prime)(x)=2\cdot(x^2+4)\cdot2x \\ g^(\prime)(x)=4x(x^2+4) \end{gathered}`

so now we can replace on the quotient rule:

`((x^2+4)^2\cdot(-8x)-4x(x^2+4)\cdot(-4x^2+16))/((x^2+4)^4)`

now we can use properties, like:

`(x^2+4)^2=x^4+8x+16`