Question

find the indicated derivative and simplify.
y for y = $\frac{5x - 3}{x^{2}+8x}$
y = □

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 5x-3$, so $u'=5$, and $v=x^{2}+8x$, so $v'=2x + 8$.

Step2: Substitute into quotient - rule formula

$y'=\frac{5(x^{2}+8x)-(5x - 3)(2x + 8)}{(x^{2}+8x)^{2}}$.

Step3: Expand the numerator

Expand $5(x^{2}+8x)=5x^{2}+40x$ and $(5x - 3)(2x + 8)=10x^{2}+40x-6x - 24=10x^{2}+34x - 24$. Then the numerator is $5x^{2}+40x-(10x^{2}+34x - 24)=5x^{2}+40x - 10x^{2}-34x + 24=-5x^{2}+6x + 24$.

Answer:

$\frac{-5x^{2}+6x + 24}{(x^{2}+8x)^{2}}$