Question

find the derivative of the following function.
s(w)=\frac{9w + 2}{sqrt{w}+6}
the derivative of s(w)=\frac{9w + 2}{sqrt{w}+6} is \frac{9w + 108sqrt{w}-2}{2sqrt{w}(sqrt{w}+6)^{2}}

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $s(w)=\frac{u(w)}{v(w)}$, then $s^\prime(w)=\frac{u^\prime(w)v(w)-u(w)v^\prime(w)}{v(w)^2}$. Here, $u(w)=9w + 2$, so $u^\prime(w)=9$, and $v(w)=\sqrt{w}+6=w^{\frac{1}{2}}+6$, so $v^\prime(w)=\frac{1}{2}w^{-\frac{1}{2}}$.

Step2: Substitute into quotient - rule formula

\[

$$\begin{align*} s^\prime(w)&=\frac{9(\sqrt{w}+6)-(9w + 2)\frac{1}{2\sqrt{w}}}{(\sqrt{w}+6)^2}\\ &=\frac{9\sqrt{w}+54-\frac{9w}{2\sqrt{w}}-\frac{2}{2\sqrt{w}}}{(\sqrt{w}+6)^2}\\ &=\frac{9\sqrt{w}+54-\frac{9}{2}\sqrt{w}-\frac{1}{\sqrt{w}}}{(\sqrt{w}+6)^2}\\ &=\frac{\frac{18\sqrt{w}-9\sqrt{w}}{2}+54-\frac{1}{\sqrt{w}}}{(\sqrt{w}+6)^2}\\ &=\frac{\frac{9\sqrt{w}}{2}+54-\frac{1}{\sqrt{w}}}{(\sqrt{w}+6)^2}\\ &=\frac{\frac{9w + 108\sqrt{w}-2}{2\sqrt{w}}}{(\sqrt{w}+6)^2}\\ &=\frac{9w + 108\sqrt{w}-2}{2\sqrt{w}(\sqrt{w}+6)^2} \end{align*}$$

\]

Answer:

$\frac{9w + 108\sqrt{w}-2}{2\sqrt{w}(\sqrt{w}+6)^2}$