$\dfrac{d}{dx}{\,\sin{(ax)}}$ rule

Formula

$\dfrac{d}{dx}{\,\sin{(ax)}}$ $\,=\,$ $a\cos{(ax)}$

Introduction

Let $a$ and $x$ be a constant and a variable respectively but the variable $x$ represents an angle in this case. The product of $a$ and $x$ is $ax$, which represents a multiple angle in mathematical form. The sine of a multiple angle $ax$ is written as $\sin{(ax)}$ mathematically.

The derivative of sine of a multiple angle $ax$ with respect to $x$ is written in the following mathematical form in calculus.

$\dfrac{d}{dx}{\,\sin{(ax)}}$

The derivative of sine of multiple angle $ax$ with respect to $x$ is equal to the product of multiple constant $a$ and the cosine of multiple angle $ax$.

$\implies$ $\dfrac{d}{dx}{\,\sin{(ax)}}$ $\,=\,$ $a \times \cos{(ax)}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{d}{dx}{\,\sin{(ax)}}$ $\,=\,$ $a\cos{(ax)}$

It is called the derivative rule for the sine of a multiple angle.

Use

It is used as a formula to find the derivative of a sine function in which multiple or submultiple angle is involved.

Example

Find $\dfrac{d}{dx}{\,\sin{(2x)}}$

In this example, $a \,=\, 2$, so, substitute it in the derivative of sine of multiple angle formula to find the derivative of $\sin{(2x)}$ with respect to $x$.

$\implies$ $\dfrac{d}{dx}{\,\sin{(2x)}}$ $\,=\,$ $2 \times \cos{(2x)}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{d}{dx}{\,\sin{(2x)}}$ $\,=\,$ $2\cos{(2x)}$