At first, when you see this problem, you might be tempted to use the quotient rule, since the expression contains a quotient. Indeed, using the quotient rule, which states that for two functions $f(x)$ and $g(x)$

\begin{align*} \frac d{dx}\bigg(\frac{f(x)}{g(x)}\bigg) = \frac{f'(x) g(x) – g'(x) f(x)}{g^2(x)}\end{align*}

We now let $f(x) = x$ and $g(x) = 2$. Then $f'(x) = 1$ and $g'(x) = 0$, which gives

\begin{align*} \dfrac d{dx} \frac x2 = \frac{2(1) – x(0)}{2^2} = \frac 2{2^2} = \boxed{\frac12}\end{align*}

This should strike you as suspicious, since the derivative of the function is a constant. In fact, we could rewrite the fraction as

\begin{align*}\dfrac12 x\end{align*}

At this point, we could use the power rule and arrive at the same result.

The takeaway is that deriving a fraction does not always require the quotient rule. If the denominator is a constant, then the quotient rule would suffice. For example, the derivative of $\dfrac x5$ would be $\dfrac15$ and the derivative of $\dfrac{3x}4$ would be $\dfrac 34$.

It is natural to ask whether this property generalizes further and, in fact, it does. For example, if you would like to differentiate the polynomial

\begin{align*} \frac{x^2}4 – \frac{3x}5 + 9\end{align*}

power rule will yield the answer

\begin{align*} 2\frac x4 – \frac35 = \frac x2 – \frac35\end{align*}

A more complicated scenario could be the derivative of $\dfrac{\ln x}2$, which would require you to know the derivative of $\ln x$. If you would like to learn the proof of the derivative of $\ln x$, you can find it here. Finding the derivative of $\dfrac{\ln x}2$ would require us to rewrite it as

\begin{align*} \frac12 \ln x\end{align*}

and at this point the derivative is clearly

\begin{align*} \frac1{2x}\end{align*}