# Implicit differentiation

1. Sep 11, 2010

### Jimmy25

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

differentiate:

yx = y / sqrt(x2 + y2)

2. Relevant equations

3. The attempt at a solution

I solved this problem by taking the ln of both sides and then solving. It seems from the context of the problem set that this was supposed to be easier than that. Am I missing a simple shortcut to the solution or did I do it the only way that is possible?

2. Sep 11, 2010

### LCKurtz

Nothing wrong with taking the ln first. You could also just differentiate it implicitly as it stands, using the quotient rule on the right. Either way should work if you slog through the details.

3. Sep 12, 2010

### Jimmy25

But how would implicitly differentiate yx?

4. Sep 12, 2010

### Whitishcube

think of it as a function inside of another function. how do we differentiate functions that have a function inside of them?

5. Sep 12, 2010

### Jimmy25

If it were a function inside another function you could use the chain rule. But in this case the variable y is being raised to the power of x another variable. If it were a number such as e raised to the power of a function you would take the derivative of e^function and then take the derivative of the function. But in this case how to take the derivative of function^function without taking a ln?

6. Sep 12, 2010

### LCKurtz

When you different a function f(x,y) with respect to x with y an implicit function of x:

$$\frac d {dx} f(x,y) = f_x + f_y y'$$

For f(x,y) = yx this gives yxln(y) + xyx-1y'.