$\dfrac{d}{dx}{\,\sin{(ax)}}$ $\,=\,$ $a\cos{(ax)}$
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.
It is used as a formula to find the derivative of a sine function in which multiple or submultiple angle is involved.
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 $x$.
$\implies$ $\dfrac{d}{dx}{\,\sin{(2x)}}$ $\,=\,$ $2 \times \cos{(2x)}$
$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{d}{dx}{\,\sin{(2x)}}$ $\,=\,$ $2\cos{(2x)}$
Learn how to prove the differentiation formula for finding the derivative of the sine of a multiple angle with respect to a variable.
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