# Derivative of Cosine

We shall prove formula for derivative of cosine function using by definition or first principle method.
Let us suppose that the function of the form $y = f\left( x \right) = \cos x$.

First we take the increment or small change in the function.

Putting the value of function $y = \cos x$ in the above equation, we get

Using formula from trigonometry, we have

Using this formula in equation (i), we get

Dividing both sides by $\Delta x$, we get

Taking limit of both sides as $\Delta x \to 0$, we have

Consider $\frac{{\Delta x}}{2} = u$, as $\Delta x \to 0$ then $u \to 0$, we get

Using the relation from limit $\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1$, we have

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

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