Question

question given the function y = (3 + x^2)/(x + 6), find dy/dx in simplified form.

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $\frac{dy}{dx}=\frac{u'v - uv'}{v^{2}}$. Here, $u = 3 + x^{2}$, so $u'=2x$, and $v=x + 6$, so $v' = 1$.

Step2: Substitute into the quotient - rule formula

$\frac{dy}{dx}=\frac{(2x)(x + 6)-(3 + x^{2})(1)}{(x + 6)^{2}}$.

Step3: Expand the numerator

Expand $(2x)(x + 6)=2x^{2}+12x$ and $(3 + x^{2})(1)=3 + x^{2}$. Then the numerator is $2x^{2}+12x-(3 + x^{2})=2x^{2}+12x - 3 - x^{2}=x^{2}+12x - 3$.

Answer:

$\frac{x^{2}+12x - 3}{(x + 6)^{2}}$