Question

find the derivative of the following function.

$f(y)=\frac{y + 6}{5\sqrt{y}+7}$

the derivative of $f(y)=\frac{y + 6}{5\sqrt{y}+7}$ is $\square$.

Explanation:

Step1: Recall the quotient rule

The quotient rule states that if \(f(y)=\frac{u(y)}{v(y)}\), then \(f^\prime(y)=\frac{u^\prime(y)v(y)-u(y)v^\prime(y)}{v(y)^2}\). Here, \(u(y) = y + 6\), so \(u^\prime(y)=1\). And \(v(y)=5y^{\frac{1}{2}}+7\), so \(v^\prime(y)=\frac{5}{2}y^{-\frac{1}{2}}\).

Step2: Substitute into the quotient rule formula

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Answer:

\(\frac{5y + 14\sqrt{y}-30}{2\sqrt{y}(5\sqrt{y}+7)^2}\)