# Implicit Differentiation

• Jan 10th 2010, 04:09 PM
suchgreatheights
Implicit Differentiation
I don't really know how to do implicit differentiation. Could you show me how to do it with the equation y-sin(xy)=x ² ? Thanks!
• Jan 10th 2010, 04:41 PM
mr fantastic
Quote:

Originally Posted by suchgreatheights
I don't really know how to do implicit differentiation. Could you show me how to do it with the equation y-sin(xy)=x ² ? Thanks!

I assume the main trouble is to get $\displaystyle \frac{d \sin (xy) }{dx}$ (if it isn't then I suggest you go back and review the topic and then attempt some easier questions). First make the substitution $\displaystyle u = xy$ and use the chain rule:

$\displaystyle \frac{d \sin (xy) }{dx} = \frac{d \sin u}{du} \cdot \frac{du}{dx} = \cos u \cdot \frac{du}{dx} = \cos (xy) \cdot \frac{d (xy)}{dx}$.

I assume you can use the product rule to get $\displaystyle \frac{d (xy)}{dx} = y + x \frac{dy}{dx}$ and then put it all together.
• Jan 10th 2010, 05:27 PM
suchgreatheights
Ok I get that but where does the first y fit in? What would be the final answer?
• Jan 10th 2010, 05:32 PM
mr fantastic
Quote:

Originally Posted by suchgreatheights
Ok I get that but where does the first y fit in? What would be the final answer?

You're meant to differentiate each term of $\displaystyle y - \sin (xy) = x^2$. I differentiated the second term for you, which I assumed was the one causing you the trouble.