# Implicit differentiation

1. Feb 24, 2009

### phantomcow2

1. The problem statement, all variables and given/known data

Find derivative of y with respect to x.

Sin(xy) = Sinx Siny

2. Relevant equations

3. The attempt at a solution

Use chain rule (product rule for inner function) to differentiate the left. Use product rule to differentiate the right and I get the following:

cos(xy)(y+xy') = (cosx siny) + (cosy' sinx)

distribute the cos(xy) on the left to get:
ycos(xy) + xy'cos(xy) = (cosx siny) + (cosy' sinx)

Rearrange to get y' on one side, everything else on the other.

ycos(xy) - cosx siny = -xy'cos(xy) + cosy' sinx

Now what? I don't understand how I'd solve for y' from here. Inverse cosine?

2. Feb 24, 2009

### Tom Mattson

Staff Emeritus
OOOOHHH, so close! You got the left side correct, and you got the first term on the right side correct. But you got the second term wrong. When you take the derivative of $\cos(y)$ with respect to $x$ you have to use the chain rule.

3. Feb 24, 2009

### Staff: Mentor

You have an error here:
d/dx(cos(y)) isn't cos(y')