Question

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

Answer 1

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

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 $${\displaystyle \frac{d}{dx}}f(g(x))$$ as, $${\displaystyle \frac{d}{dx}}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$$. Let $$f(x)=\sin x$$ and $$g(x)=ax$$, then
$${\displaystyle \frac{d}{dx}}\sin (ax)={\displaystyle \frac{d\sin (ax)}{d(ax)}}\times {\displaystyle \frac{d(ax)}{dx}}$$
Since we know that the derivative of $$\sin (x)$$ with respect to $$x$$ is given by $${\displaystyle \frac{d}{dx}}\sin (x)=\cos x$$ and derivative of $$x$$ with respect to $$x$$ is equal to $$1$$, we proceed to further steps using above mentioned formulas as,
$$\begin{gathered} \Rightarrow {\displaystyle \frac{d}{dx}}\sin (ax)={\displaystyle \frac{d\sin (ax)}{d(ax)}}\times {\displaystyle \frac{adx}{dx}} \\ \Rightarrow {\displaystyle \frac{d}{dx}}\sin (ax)=\cos (ax)\times a \\ \Rightarrow {\displaystyle \frac{d}{dx}}\sin (ax)=a\cos (ax) \end{gathered}$$
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 $$x$$.