Question

How do you find the derivative of $y=\sin \left( x\cos x \right)$ using the chain rule?

Answer 1

We will first see the formula given by the chain rule and its definition. Then we will see the composition of functions in the given function. We will identify this composition of functions with the chain rule. Then we will use the chain rule to find the derivative of the given function. We will also use the product rule for derivatives to find the derivative of one of the functions in the composition.

Complete step-by-step solution:
The chain rule is defined as the rule for computing derivative of a composition of functions. The chain rule is given as,
${{\left( f\circ g \right)}^{\prime }}=\left( {f}'\circ g \right)\cdot {g}'$
The given function is $y=\sin \left( x\cos x \right)$. So, we have $f\left( x \right)=\sin x$ and $g\left( x \right)=x\cos x$. Let us find the derivative of the function $g\left( x \right)$, We can see that this function is a product of two functions. These two functions are $x$ and $\cos x$. We have the product rule for derivatives which is given as,
$\dfrac{d}{dx}\left( uv \right)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$
Using this rule, we find the derivative of the function $g\left( x \right)$ in the following manner,
$\begin{aligned} & \dfrac{d}{dx}\left( g\left( x \right) \right)=\dfrac{d}{dx}\left( x\cos x \right) \\\ & \therefore \dfrac{d}{dx}\left( g\left( x \right) \right)=x\dfrac{d}{dx}\left( \cos x \right)+\cos x\dfrac{d}{dx}\left( x \right) \\\ \end{aligned}$
We know that the derivative of the function $x$ is 1 and the derivative of the cosine function is $-\sin x$. Substituting these values in the above equation, we get
$\begin{aligned} & \dfrac{d}{dx}\left( g\left( x \right) \right)=x\left( -\sin x \right)+\cos x \\\ & \therefore {g}'\left( x \right)=-x\sin x+\cos x \\\ \end{aligned}$
We know that the derivative of the sine function is $\cos x$. So, using the chain rule, we have the following,
$\dfrac{dy}{dx}=\dfrac{d}{dx}\left( \sin \left( x\cos x \right) \right)\cdot \dfrac{d}{dx}\left( x\cos x \right)$
Substituting the values of the derivatives, we get the required derivative as
$\dfrac{dy}{dx}=\cos \left( x\cos x \right)\cdot \left( \cos x-x\sin x \right)$

Note: The chain rule is a very important concept in computing derivatives. We should be familiar with the derivatives of the standard functions for such types of questions. There are multiple rules for derivatives of addition, subtraction, product etc of functions. These rules are useful in computing the derivatives of such functions.