find the derivative of f(x) = x^4 / (x^2 + 2x + 4). f(x) =

Answer:

$\frac{4x^{3}(x^{2}+2x + 4)-x^{4}(2x + 2)}{(x^{2}+2x + 4)^{2}}$

Explanation:

Step1: Recall quotient - rule

If $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here $u = x^{4}$, $v=x^{2}+2x + 4$.

Step2: Find $u^\prime$

Using the power - rule $(x^{n})^\prime=nx^{n - 1}$, for $u = x^{4}$, $u^\prime=4x^{3}$.

Step3: Find $v^\prime$

For $v=x^{2}+2x + 4$, $v^\prime=(x^{2})^\prime+(2x)^\prime+(4)^\prime=2x + 2$.

Step4: Apply quotient - rule

$f^\prime(x)=\frac{u^\prime v - uv^\prime}{v^{2}}=\frac{4x^{3}(x^{2}+2x + 4)-x^{4}(2x + 2)}{(x^{2}+2x + 4)^{2}}$.