Question

given the function $f(x)=\frac{1 + 3x^{2}}{5x^{2}+3}$, find $f(x)$ in simplified form.

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 1 + 3x^{2}$, $u'=6x$, $v = 5x^{2}+3$, and $v' = 10x$.

Step2: Apply quotient - rule

$f'(x)=\frac{(6x)(5x^{2}+3)-(1 + 3x^{2})(10x)}{(5x^{2}+3)^{2}}$.

Step3: Expand the numerator

Expand $(6x)(5x^{2}+3)=30x^{3}+18x$ and $(1 + 3x^{2})(10x)=10x+30x^{3}$. Then the numerator is $30x^{3}+18x-(10x + 30x^{3})$.

Step4: Simplify the numerator

$30x^{3}+18x - 10x-30x^{3}=8x$.

Answer:

$\frac{8x}{(5x^{2}+3)^{2}}$