Question

Find dy/dx if ln(xy)= x+y?

Answer 1

You need to use implicit differentiation.

Answer 2

Implicit differentiation.

Answer 3

Do you need help in figuring out how to use the process of implicit differentiation?

Answer 4

Yes please

Answer 5

The first step in implicit differentiation, is to simply take the derivative, pretending the variables are actually functions. So in this case, you would write it as\[\begin{aligned}\frac{d}{dx}(\ln(xy)=x+y)&\leadsto\frac{d}{dx}\ln(xy)=\frac{d}{dx}(x+y)\\
&\leadsto\frac{d}{dx}\ln(xy)=\frac{d}{dx}x+\frac{d}{dx}y
\end{aligned}\]The right side is easy since \(dx/dx=1\) and we want to solve for \(dy/dx\). As for the left side, we'll need to use the chain rule and product rule.

Is this making sense so far?

Answer 6

Yes

Answer 7

Would the answer be xy-y/x-xy?

Answer 8

That looks correct to me. Assuming there are parentheses so that it can be written as:\[\frac{xy-y}{x-xy}\]

Answer 9

Okay! Thank you so much! :)

Answer 10

You're welcome.