How do I handle square roots in partial derivatives?

[Arden1528]

All right, I know that partial derv. are not all that hard. All you do in let's say f(x)= x^2+2yx+3y is find the dervitives of x while you treat y as a constent, and vice versa. But I keep running into problems having square roots. I hat these things.
One example could be f(x)= sqr(20-x^2-7y^2). I know yo treat the y as a constent, but I am horrible with my derv. Any help would be appreciated.

 

[StephenPrivitera]

If you hate square roots so much, would you be more comfortable with implicit differentiation? Can you easily take the deriviatve of g(x)=20-x²-7y²? Sure you can. Now, g(x)=[f(x)]² and 2ff'=g' so f'=g'/(2f) which agrees with what you know about square roots. You know f, so all you need is g', which is easy to find.
g'=-2x
So the partial derivative of f wrt x is,
f'=(-2x)/(2f(x))=-x/f(x)

Please forgive the sloppy notation.

 

[Hurkyl]

The other way is to apply the chain rule to the power rule:

(d/dx) (u^n) = n u^(n-1) (du/dx)

then just plug in the right things for u and n and finish the rest of the work

 

[Soroban]

Hello, Arden1528!

You said, "I keep running into problems having square roots.
One example could be f(x,y)= sqr(20 - x^2 - 7y^2)"

Just where is your difficulty?
You know how to take partial derivatives.
You know that a square root is a one-half power.
And you know the Chain Rule.

f(x,y) = (20 - x² - 7y²)^1/2

Hence, f_x = (1/2)(20 - x² - 7y²)^-1/2(-2x) = -x/(20 - x² - 7y²)^1/2

And, f_y = {1/2)(20 - x² - 7y²)^-1/2(-14y) = -7y/(20 - x² - 7y²)^1/2

 

[Arden1528]

I did not even think about just putting the problems to the (1/2) power. I seem to forget the things that would make these problems easier. But thanks for all the help. I feel like I just forgot a really elementry operation...thanks