Using the definition of the partial derivative

[cchase88]

I need some help with this partial derivative. I can do it by rules, but when I try and do it out using the definition of the partial derivative, I run into problems.

Homework Statement

Find the partial derivative of sqrt[x]y^2 - 4xy with respect to x

The Attempt at a Solution

Going through while holding a variable constant and using basic derivative rules, I was able to get:

(y^2) / 2sqrt[x] - 4y

 

[lanedance]

hi
As long as that is
(y^2) / (2.sqrt[x]) - 4y
looks good to me...

 

[cchase88]

I got that answer by using derivative rules, and not by the definition. When you plug in x+h for x you get sqrt[x+h].. and that's where I'm getting stuck.

I come out getting this:

(4hy + (sqrt[x+h] - sqrt[x])y^2)/h

 

[Mark44]

I got that answer by using derivative rules, and not by the definition. When you plug in x+h for x you get sqrt[x+h].. and that's where I'm getting stuck.

I come out getting this:

(4hy + (sqrt[x+h] - sqrt[x])y^2)/h

It's getting close to bedtime, so I don't have time to check your work up to this point. Assuming that it's correct up to this point, split the expression above into the sum of two terms, like so:
4hy/h + (sqrt(x + h) - sqrt(x))*y
$$\frac{4hy}{h} + \frac{y^2(\sqrt{x + h} - \sqrt{x})}{h}$$

It'll be easy to take the limit as h goes to zero for the term on the left, but the one on the right needs a trick, namely to multiply it (the quotient on the right) by 1 in the form of
$$\frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}}$$

That should give you something that you can take the limit on.