## What is the derivative of x sin x?

Answer:

\(\frac{d y}{d x}=x \cos x+\sin x\)

Explanation:

We have:

Which is the product of two functions, and so we apply the Product Rule for Differentiation:

\(\frac{d}{d x}(u v)=u \frac{d v}{d x}+\frac{d u}{d x} v, \text { or, }(u v)^{\prime}=(d u) v+u(d v)\).

I was taught to remember the rule in words; “The first times the derivative of the second plus the derivative of the first times the second “.

So with \(y=x \sin x\)

Then:

Gives us:

\(\frac{d}{d x}(x \sin x)=(x)(\cos x)+(1)(\sin x)\)

\(∴ \frac{d y}{d x}=x \cos x+\sin x\)

If you are new to Calculus then explicitly substituting \(u\) and \(v\) can be quite helpful, but with practice these steps can be omitted, and the product rule can be applied as we write out the solution.