To find the derivative of \( x\sin x \) we will need to use the Product Rule, Power Rule on \( x \), and the derivative of \( \sin x \).
By the Product Rule, \( (fg)’=f’g+g’f \).
By the Power Rule, the derivative of \( x \) is equal to \( 1 \).
The derivative of \( \sin x \) is equal to \( \cos x \).
Now, let’s put this all together to compute the derivative
\( \dfrac{d}{dx}(x\sin x)=(x)’\sin x + x\cdot\cos x = \boxed{\sin x + x \cos x} \)
Note that here we cannot differentiate \(x\) and \(\sin x\) individually, which would be a common mistake. When you have a product of terms, please use the product formula.
