Question

What is the derivative of $\dfrac{d}{{dx}}\cos \left( {ax} \right)$?

Answer 1

Hint : Here, in the given question, we are given a trigonometric function $\cos \left( {ax} \right)$. We have to find the derivative of the given function with respect to $x$. Since we have to differentiate with respect to $x$ but the function is $ax$. So, to find the derivative of the given trigonometric function we will apply the chain rule. Here two functions on which chain rule will be applied will be $\cos \left( x \right)$ and $ax$.
Formula used
$\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right).g'\left( x \right)$
In words: differentiate the outside function, and then multiply by the derivative of the inside function.

Complete step-by-step answer :
To solve the given trigonometric function we will apply chain rule. According to chain rule, if we are given two functions $f$ and $g$ then we find $\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right)$ as,
$\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right).g'\left( x \right)$
Let $f\left( x \right) = \cos x$ and $g\left( x \right) = ax$, then
$\Rightarrow \dfrac{d}{{dx}}\cos \left( {ax} \right) = \dfrac{d}{{d\left( {ax} \right)}}\cos \left( {ax} \right) \times \dfrac{d}{{dx}}\left( {ax} \right)$
$\Rightarrow \dfrac{d}{{dx}}\cos \left( {ax} \right) = \dfrac{d}{{d\left( {ax} \right)}}\cos \left( {ax} \right) \times \dfrac{{adx}}{{dx}}$
Since we know that the derivative of $\cos \left( x \right)$ with respect to $x$ is given by $\dfrac{d}{{dx}}\cos \left( x \right) = - \sin x$ and derivative of $x$ with respect to $x$ is $1$. Therefore, we get
$\Rightarrow \dfrac{d}{{dx}}\cos \left( {ax} \right) = - \sin \left( {ax} \right) \times a$
$\Rightarrow \dfrac{d}{{dx}}\cos \left( {ax} \right) = - a\sin \left( {ax} \right)$
Hence, the derivative of the given trigonometric function $\dfrac{d}{{dx}}\cos \left( {ax} \right)$ is $- a\sin \left( {ax} \right)$.
So, the correct answer is “ $- a\sin \left( {ax} \right)$”.

Note : Differentiation is a process of finding the rate of change of a function. Here, we have found the rate of change of $\cos \left( {ax} \right)$ with respect to $x$. Remember that the derivative of a constant with respect to any variable is zero. Also, the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. The given question can also be solved by using the first principle of differentiation. First principle: Suppose $f$ is a real function and $x$ is a point in its domain of definition. The derivative of $f$ at $x$ is defined by $\mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$ provided this limit exists.