Partial Derivative of x^y: How to Find the First Partial Derivatives?

[cookiesyum]

Homework Statement

Find the first partial derivatives of:

1. f(x,y) = x^y
2. u = x^(y/z)

Homework Equations

The Attempt at a Solution

f_x = y*x^(y-1)
f_y = lnx?

u_x = (y/z)*x^((y/z)-1)
u_y = lnx/z?
u_z = ylnx/z?

I'm not really sure how to do these right. =/ I would really appreciate any help.

 

[jambaugh]

Your f_x is right. Your f_y is not. Look at x as a constant in this one and look up the derivative of an exponential of arbitrary base formula.

Your u_x is right.
Your u_y again should be treated as an exponential function base x.
Your u_z should as well with an additional application of the chain rule.

 

[cookiesyum]

Thank you!

 

[Bacat]

Don't forget to use the chain-rule.

For the y derivative of x^y:

Let x = k, a constant.

$$f(y) = k^y$$

Natural log of both sides gives:

$$ln(f(y)) = ln(k^y)$$

$$ln(f(y)) = yln(k)$$

Differentiating...

$$f'(y)/f(y) = ln(k)$$

$$f'(y) = f(y)ln(k)$$

Since $$f(y) = k^y$$, you now have:

$$f'(y) = ln(k)k^y$$

Substituting for x...

$$f_y = ln(x)x^y$$