Question

What is the derivative of sin(xy)? Please show your individual steps if possible.

Answer 1

Do you want to find the partial derivatives?

Answer 2

Yes, I think with respect to d/dx. So d/dx[sin(xy)], if that makes sense.

Answer 3

here it is but its backwards

Answer 4

lol it's backwards taylor! XD

Answer 5

As sam asked, derivative with respect to x, \(\Large \dfrac{d}{dx}\)

or partial derivative with respect to x, \(\Large \dfrac{\partial}{\partial x}\)

Whichhhh? D:

Answer 6

Oh, I meant derivative with respect to x (d/dx), sorry!

Answer 7

Ok so we take the derivative of the outer function, \(\large \sin(stuff)\) in this case :)
which gives us \(\large \cos(stuff)\) right?

\[\Large \left[\sin(xy)\right]'\quad=\quad \left[\cos(xy)\right]\color{royalblue}{(xy)'}\]

Answer 8

The chain rule tells us to multiply by the derivative of the "stuff" inside.

Answer 9

So we still need to take the derivative of this blue portion.

Answer 10

We'll apply the product rule for the blue part.\[\Large \left[\cos(xy)\right]\left[\color{royalblue}{(x)'}y+x\color{royalblue}{(y)'}\right]\]

Answer 11

\[\Large (x)'\quad=\quad?\]Derivative of x, with respect to x? :)

Answer 12

(x)'=1, so it'd be ycos(xy) + xy'cos(xy) after distributing (?).

Answer 13

Yes, if you want to distribute :) that looks correct!

Answer 14

Ok, thank you all for the help!