Logarithmic Differentiation with Respect to y

[captainjack2000]

Homework Statement

I am trying to differentiate log((x+(x^2+y^2)^0.5)/(-x+(x^2+y^2)^0.5)) with respect to y

Homework Equations

I know that d/dx of ln(x) = 1/x but i am getting really confused when it comes to differentiating wrt to y?

Can I have some help please!

The Attempt at a Solution

 

[rock.freak667]

Use the chain rule

$$\frac{d}{dy}= \frac{dx}{dy}*\frac{d}{dx}$$

So if you had to differentiate ln(x) w.r.t y

$$\frac{d}{dy}(lnx)= \frac{dx}{dy}*\frac{d}{dx}(lnx) = \frac{1}{x}*\frac{dx}{dy}$$

 

[djeitnstine]

The argument is $$\frac{x+\sqrt{x^2+y^2}}{-x+\sqrt{x^2+y^2}}$$

So when differentiating with respect to y it looks like $$\frac{a+\sqrt{a+y^2}}{-a+\sqrt{a+y^2}}$$

where x is just treated as some constant a

So you just use the chain rule for that derivative

Does that look a little less confusing?