Partial Derivative of x^y?

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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.
 

Answers and Replies

  • #2
jambaugh
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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.
 
  • #3
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Thank you!
 
  • #4
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Don't forget to use the chain-rule.

For the y derivative of x^y:

Let x = k, a constant.

[tex]f(y) = k^y[/tex]

Natural log of both sides gives:

[tex]ln(f(y)) = ln(k^y)[/tex]

[tex]ln(f(y)) = yln(k)[/tex]

Differentiating...

[tex]f'(y)/f(y) = ln(k)[/tex]

[tex]f'(y) = f(y)ln(k)[/tex]

Since [tex]f(y) = k^y[/tex], you now have:

[tex]f'(y) = ln(k)k^y[/tex]

Substituting for x...

[tex]f_y = ln(x)x^y[/tex]
 

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