Q:

derivative of sqrt(5/(x+7))

Why is it that:
$$\frac{d}{dx}\sqrt{\frac{5}{x+7}} = -\frac{\sqrt{5}}2\frac{1}{(x+7)^{3/2}}$$ (image)
???
My attempt:

It seems that somehow you end up adding 1 to 1/2 to get 3/2 in the exponent.
But why?

A:

You may write
$$
\begin{align}
\left(\sqrt{\frac{5}{x+7}}\: \right)'&=\sqrt{5}\left(\frac{1}{\sqrt{{x+7}}}\: \right)'\\\\&=\sqrt{5}\left((x+7)^{\large-\frac12}\right)'\\\\
&=\sqrt{5}\:\left(-\frac12\right)(x+7)^{\large-\frac12-1}\\\\
&=-\frac{\sqrt{5}}2\frac{1}{(x+7)^{3/2}}.
\end{align}
$$

