differentiate the function. y = 2/π + 4/(x² + 6) y = □

Explanation:

Step1: Recall derivative rules

The derivative of a constant is 0, and for a function of the form $\frac{u}{v}$, the quotient - rule is $(\frac{u}{v})'=\frac{u'v - uv'}{v^{2}}$. The derivative of a sum $y = f(x)+g(x)$ is $y'=f'(x)+g'(x)$. The constant $\frac{2}{\pi}$ is a constant, so its derivative is 0. Let $u = 4$ and $v=x^{2}+6$ for the second term.

Step2: Differentiate the second - term

First, find $u'$ and $v'$. Since $u = 4$, then $u'=0$. Since $v=x^{2}+6$, then $v' = 2x$. Using the quotient - rule $(\frac{u}{v})'=\frac{u'v - uv'}{v^{2}}$, we have $(\frac{4}{x^{2}+6})'=\frac{0\times(x^{2}+6)-4\times2x}{(x^{2}+6)^{2}}=\frac{-8x}{(x^{2}+6)^{2}}$.

Step3: Find the derivative of the whole function

Since $y=\frac{2}{\pi}+\frac{4}{x^{2}+6}$, and the derivative of $\frac{2}{\pi}$ is 0 and the derivative of $\frac{4}{x^{2}+6}$ is $\frac{-8x}{(x^{2}+6)^{2}}$, then $y'=0+\frac{-8x}{(x^{2}+6)^{2}}$.

Answer:

$\frac{-8x}{(x^{2}+6)^{2}}$