# Derivative of Cosine Square Root of X

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

Consider the function of the form

We can prove this with the help of definition of differentiation, we have

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

Using 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

Differentiation 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