Question

How do you differentiate $f\left( x \right)=2x\sin x$?

Answer 1

Assume the given function as $f\left( x \right)$. Consider $f\left( x \right)$ as the product of an algebraic function and a trigonometric function. Now, apply the product of differentiation given as: - $\dfrac{d\left( u\times v \right)}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$. Here, consider, u = x and $v=\sin x$. Use the formula: - $\dfrac{d\sin x}{dx}=\cos x$ to simplify the derivative and get the answer.

Complete step by step answer:
Here, we have been provided with the function $2x\sin x$ and we are asked to differentiate it. Let us assume the given function as $f\left( x \right)$. So, we have,
$\Rightarrow f\left( x \right)=2x\sin x$
Now, we can assume the given function as the product of an algebraic function (x) and a trigonometric function $\left( \sin x \right)$, which are multiplied to a constant 2. So, we have,
$\Rightarrow f\left( x \right)=2\times x\times \sin x$
Let us assume x as ‘u’ and $\sin x$ as ‘v’. So, we have,
$\Rightarrow f\left( x \right)=2\times u\times v$
Differentiating both the sides with respect to x, we get,
$\Rightarrow \dfrac{df\left( x \right)}{dx}=\dfrac{d\left[ 2\left( u\times v \right) \right]}{dx}$
Since, 2 is a constant so it can be taken out of the derivative, so we get,
$\Rightarrow f'\left( x \right)=2\dfrac{d\left( u\times v \right)}{dx}$
Now, applying the product rule of differentiation given as: - $\dfrac{d\left( u\times v \right)}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$, we get,
$\Rightarrow f'\left( x \right)=2\left[ u\dfrac{dv}{dx}+v\dfrac{du}{dx} \right]$
Substituting the assumed values of u and v, we get,
$\Rightarrow f'\left( x \right)=2\left[ x\dfrac{d\sin x}{dx}+\sin x\dfrac{dx}{dx} \right]$
We know that $\dfrac{d\sin x}{dx}=\cos x$, so we have,

& \Rightarrow f'\left( x \right)=2\left[ x\cos x+\sin x\times 1 \right] \\\ & \Rightarrow f'\left( x \right)=2\left( x\cos x+\sin x \right) \\\ \end{aligned}$$ Hence, the above relation is our answer. **Note:** One may note that whenever we are asked to differentiate a product of two or more functions we apply the product rule. You must remember all the basic rules and formulas of differentiation like: - the product rule, chain rule, $$\dfrac{u}{v}$$ rule etc. as they make our question easy to solve.