# Derivative of the square root of xy

1. Jan 11, 2009

### brambleberry

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

What is the deriv. of the square root of (xy)?

2. Relevant equations

3. The attempt at a solution

I used the chain rule:

(1/2)(xy)^(-1/2) times (y + x(dy/dx))

i am unsure on how to distribute this correctly

2. Jan 11, 2009

### HallsofIvy

Staff Emeritus
The derivative of $(xy)^{1/2}$ with respect to x, and y is a function of x?

If that is the question, then yes that is correct. I don't know what you mean "how to distribute this correctly". The distributive law is the distributive law: a(b+ c)= ab+ ac.
Is it the half powers that concern you? $(xy)^{1/2}x= (x^{1/2})(x)(y^{1/2}= x^{3/2}y^{1/2}$ and $(xy)^{1/2}y= (x^{1/2})(y^{1/2})y= x^{1/2}y^{3/2}$.

$(1/2)(xy)^{1/2}[y+ x dy/dx]= (1/2)x^{1/2}y^{3/2}+ x^{3/2}y^{1/2} dy/dx$

3. Jan 11, 2009

### NoMoreExams

Are you trying to do implicit differentiation? If so treat

$$\sqrt{xy} = \sqrt{x}\sqrt{y}$$

Then use the product rule, just remember when you differentiate $$\sqrt{y}$$ to multiply by y'.