Question

What is the derivative of $\sin (ax)$, where $a$ is a constant ?

Answer 1

Here we are given a trigonometric function$\sin (ax)$. We have to find the derivative of this function. Since we have to derivate it with respect to $x$ but function is $ax$. So to derivate this trigonometric function we apply chain rule. Here two functions on which chain rule will be applied will be $\sin (x)$ and $ax$.

Formula used: Derivative of $\sin (x)$ with respect to $x$ is given by $\dfrac{d}{{dx}}\sin (x) = \cos x$.
Derivative of $x$ with respect to $x$ is given by $\dfrac{d}{{dx}}x = 1$.
To derivate the trigonometric function having if we are given two functions $f$ and $g$ with respect to $x$ we apply chain rule as
$\dfrac{d}{{dx}}f(g(x)) = f'(g(x)) \cdot g'(x)$.

Complete step-by-step answer:
To solve the given trigonometric function we have to apply chain rule. According to chain rule, if we are given two functions $f$ and $g$ then we find $\dfrac{d}{{dx}}f(g(x))$ as, $\dfrac{d}{{dx}}f(g(x)) = f'(g(x)) \cdot g'(x)$. Let $f(x) = \sin x$ and $g(x) = ax$, then
$\dfrac{d}{{dx}}\sin (ax) = \dfrac{{d\sin (ax)}}{{d(ax)}} \times \dfrac{{d(ax)}}{{dx}}$
Since we know that the derivative of $\sin (x)$ with respect to $x$ is given by $\dfrac{d}{{dx}}\sin (x) = \cos x$ and derivative of $x$ with respect to $x$ is equal to $1$, we proceed to further steps using above mentioned formulas as,

$$\Rightarrow \dfrac{d}{{dx}}\sin (ax) = \dfrac{{d\sin (ax)}}{{d(ax)}} \times \dfrac{{adx}}{{dx}} \\\ \Rightarrow \dfrac{d}{{dx}}\sin (ax) = \cos (ax) \times a \\\ \Rightarrow \dfrac{d}{{dx}}\sin (ax) = a\cos (ax) \\\$$

Hence, the derivative of the given trigonometric function comes out to be $a\cos (ax)$.

Note: This is to note that derivative of constant with respect to any variable is zero. Chain rule we have used here is a way to differentiate composite functions. Differentiation is a method or process of finding the derivative or rate of change of a function. Here we have found the rate of change of $\sin (ax)$ with respect to $x$.