Question

How do you differentiate $f(x)={{x}^{2}}\cos x$ using product rule?

Answer 1

The product rule is a rule for differentiating expressions in which one function is multiplied by another function. The rule follows from the limit definition of derivative
and is given by $\dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$ if $y=uv$ where $\dfrac{dy}{dx}$ means derivative of y with respect to x, $\dfrac{dv}{dx}$ means derivative of v with respect to x and $\dfrac{du}{dx}$ means derivative of u with respect to x. The above formula is applicable only if u and v are differentiable.

Complete step by step answer:
As per the given question, we have to differentiate the given function using the product rule. Here, the given function to be differentiated is $f(x)={{x}^{2}}\cos x$.
Now, let $y=f(x)$ then $u={{x}^{2}}$ and $v=\cos x$. We know that, according to power rule,
derivative of $a{{x}^{n}}$ is given by $\dfrac{d}{dx}(a{{x}^{n}})=(na){{x}^{n-1}}$ and the derivative of trigonometric function $\cos x$ is $\dfrac{d}{dx}(\cos x)=-\sin x$.
The derivative of function u: -
By comparing the function ‘u’ with the function of power rule, we get the coefficient $a=2$ and power of x, $n=1$. Then,
$\Rightarrow \dfrac{du}{dx}=2x$ ($\Leftarrow$according to power rule)
The derivative of function v: -
By comparing the function ‘v’ with the trigonometric function, function ‘v’ is the same as the above trigonometric function. Then,
\Rightarrow $$$$\dfrac{d}{dx}(\cos x)=-\sin x ($\Leftarrow$from derivative of logarithmic function)
Now substituting the above derivatives in product rule, we get
\Rightarrow $$$$\dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}

& \Rightarrow \dfrac{dy}{dx}={{x}^{2}}\dfrac{d(\cos x)}{dx}+\cos x\dfrac{d({{x}^{2}})}{dx} \\\ & \Rightarrow \dfrac{dy}{dx}={{x}^{2}}(-\sin x)+\cos x(2x) \\\ \end{aligned}$$ On solving the above equation, we get $$\Rightarrow \dfrac{dy}{dx}=2x\cos x-{{x}^{2}}\sin x=x(2\cos x-x\sin x)$$ $$\Rightarrow \dfrac{dy}{dx}=\dfrac{d(f(x)}{dx}=x(2\cos x-x\sin x)$$ **$$\therefore $$ The derivative of the given function $$f(x)={{x}^{2}}\cos x$$ is $$x(2\cos x-x\sin x)$$.** **Note:** In order to solve these types of problems, we must have enough knowledge about derivatives of basic functions. The product rule is nothing but take the derivative of v multiplied by u and add v multiplied by the derivative of u. We have the product rule for 3 functions also which is given by $$\dfrac{dy}{dx}=\dfrac{df(x)}{dx}g(x)h(x)+f(x)\dfrac{dg\left( x \right)}{dx}h(x)+f(x)g(x)\dfrac{dh(x)}{dx}$$ if $$y=f(x)g(x)h(x)$$.