find the critical points of the following function.\n\n$f(x)=\\frac{4x}{x^{2}+16}$\n\nwhat is the derivative…

find the critical points of the following function.\n\n$f(x)=\\frac{4x}{x^{2}+16}$\n\nwhat is the derivative of $f(x)=\\frac{4x}{x^{2}+16}$?\n\n$f(x)=\\square$

find the critical points of the following function.\n\n$f(x)=\\frac{4x}{x^{2}+16}$\n\nwhat is the derivative of $f(x)=\\frac{4x}{x^{2}+16}$?\n\n$f(x)=\\square$

Answer

Explanation:

Step1: Apply quotient rule

Quotient rule: ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^2}). Here (u = 4x), (u^\prime=4); (v=x^{2}+16), (v^\prime = 2x).

Step2: Substitute into quotient rule formula

[ \begin{align*} f^\prime(x)&=\frac{4(x^{2}+16)-4x\times(2x)}{(x^{2}+16)^{2}}\ &=\frac{4x^{2}+64 - 8x^{2}}{(x^{2}+16)^{2}}\ &=\frac{-4x^{2}+64}{(x^{2}+16)^{2}} \end{align*} ]

Answer:

(\frac{-4x^{2}+64}{(x^{2}+16)^{2}})