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

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Answer:

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