Recall the power rule for taking derivatives of polynomials:
$$\dfrac{d}{dx} x^n = nx^{n – 1}$$
The power rule works for all real values of $n$, not jus the integers. We may apply this rule to derive $\sqrt x = x^{\frac12}$. Using the rule above with $n =\frac12$ we get
$$\dfrac{d}{dx} x^{\frac12} = \frac12 x^{-\frac12}$$
Usually, the derivative of $\sqrt x$ is written in the form
$$\boxed{\dfrac1{2\sqrt x}}$$
This form is often preferable when taking derivatives of more complicated functions. Try to guess the derivative of
$$\sqrt{x+1}$$
If you guessed
$$\dfrac1{2\sqrt{x+1}}$$
you have good intuition! However, this trick will not work when functions under the square root are more complicated. For example, the derivative of
$$\sqrt{3x + 1}$$
is not equal to
$$\frac1{2\sqrt{3x +1}}$$
The reason is that the function under the square root is not “simple”. That is, you must use the chain rule when taking the derivative in this case. However, there is a quick shortcut for all such cases. The correct answer to the problem is
$$\frac3{2\sqrt{3x +1}}$$
Can you guess the derivative of
$$\sqrt{x^2 + 1}$$
If you guessed
$$\frac{x}{\sqrt{x^2+1}}$$
then you have great intuition. In general, to differentiate $\sqrt{f(x)}$, where $f(x)$ is some function, the result will always be
$$\frac{f'(x)}{2\sqrt{f(x)}}$$
Try using this shortcut to differentiate
$$\sqrt{x^3 + 3}$$
If your answer was
$$\dfrac{3x^2}{2\sqrt{x^3+3}}$$
then you were correct! You are using the shortcut correctly, great job! If you want to try another example, derive
$$\sqrt{\sin x}$$
The correct answer is
$$\frac{\cos x}{2\sqrt{\sin x}}$$
