Question

What is the derivative of `xsinx`?

Answer 1

`dy/dx = xcosx+sinx`

Explanation:

We have:

`y = xsin x`

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

`d/dx(uv)=u(dv)/dx+(du)/dxv`, or, `(uv)' = (du)v + u(dv)`

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 = xsinx`;

`{ ("Let", u = x, => (du)/dx = 1), ("And" ,v = sinx, => (dv)/dx = cosx ) :}`

Then:

`d/dx(uv)=u(dv)/dx + (du)/dxv`

Gives us:

`d/dx( xsinx) = (x)(cosx)+(1)(sinx)`
`:. dy/dx = xcosx+sinx`

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.