 rheisf

2021-12-29

What is the derivative of $x\mathrm{sin}x$ Bernard Lacey

Expert

$\frac{dy}{dx}=x\mathrm{cos}x+\mathrm{sin}x$
Explanation:
We have:
$y=x\mathrm{sin}x$
Which is the product of two functions, and so we apply the Product Rule for Differentiation:

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 $y=x\mathrm{sin}x$

Then:
$\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+\frac{du}{dx}v$
Gives us:
$\frac{d}{dx}\left(x\mathrm{sin}x\right)=\left(x\right)\left(\mathrm{cos}x\right)+\left(1\right)\left(\mathrm{sin}x\right)$
$\therefore \frac{dy}{dx}=x\mathrm{cos}x+\mathrm{sin}x$
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. Maricela Alarcon

Expert

This is a function which is in the form,
$y=f\left(x\right)g\left(x\right)$
Its Vasquez

Expert