Question

find the derivative of the function h(r), below. it may be to your advantage to simplify first. h(r)=\frac{r^{6}}{15r + 5} h(r)=

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $h(r)=\frac{f(r)}{g(r)}$, then $h^{\prime}(r)=\frac{f^{\prime}(r)g(r)-f(r)g^{\prime}(r)}{g(r)^2}$. Here, $f(r)=r^{6}$, so $f^{\prime}(r) = 6r^{5}$; and $g(r)=15r + 5$, so $g^{\prime}(r)=15$.

Step2: Substitute into quotient - rule formula

$h^{\prime}(r)=\frac{6r^{5}(15r + 5)-r^{6}\times15}{(15r + 5)^{2}}$.

Step3: Expand the numerator

Expand $6r^{5}(15r + 5)-15r^{6}$:

$$
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$$

So, $h^{\prime}(r)=\frac{75r^{6}+30r^{5}}{(15r + 5)^{2}}$.

Answer:

$\frac{75r^{6}+30r^{5}}{(15r + 5)^{2}}$