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 \[\dfrac{d}{{dx}}\sin (x) = \cos x\].
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 \[x\].