# Derivative of Cosine Square Root of X

In trigonometric differentiation, most of the examples are based on the sine square roots function. We will discuss the derivative of cosine square root of the $x$ function and its related examples in detail. It can be proved by the definition of differentiation.

Consider the function of the form

We can prove this with the help of the definition of differentiation:

Putting the value of the function in equation (i), we get

Using the formula from trigonometry

Now consider the relation

Consider $\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2} = u$, as $\Delta x \to 0$, then $u \to 0$

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}\cos \sqrt x = - \frac{{\sin \sqrt x }}{{2\sqrt x }}$, we get