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

In summary, to find the first partial derivatives of the given functions, use the chain rule and the derivative of an exponential function. For f(x,y) = x^y, the partial derivatives are f_x = y*x^(y-1) and f_y = lnx*x^y. For u = x^(y/z), the partial derivatives are u_x = (y/z)*x^((y/z)-1), u_y = lnx*x^(y/z), and u_z = -(y/z^2)*x^(y/z).

## Homework Statement

Find the first partial derivatives of:

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

## 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.

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_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.

Thank you!

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...

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

## 1. What is the partial derivative of x^y with respect to x?

The partial derivative of x^y with respect to x is yx^(y-1), where y is a constant. This means that when taking the partial derivative of x^y, the exponent y remains constant, and the coefficient becomes y times x^(y-1).

## 2. How is the partial derivative of x^y different from the regular derivative?

The partial derivative of x^y is different from the regular derivative because it only considers the change in one variable while holding the other variables constant. In contrast, the regular derivative considers the change in all variables simultaneously.

## 3. What is the purpose of finding the partial derivative of x^y?

The partial derivative of x^y is used to determine the rate of change of x^y with respect to a particular variable, while holding all other variables constant. This is useful in various fields of science, such as physics, engineering, and economics.

## 4. How do you find the partial derivative of x^y?

To find the partial derivative of x^y, you need to use the power rule, where you bring down the exponent as a coefficient and subtract 1 from the exponent. For example, the partial derivative of x^3 would be 3x^2, and the partial derivative of x^4 would be 4x^3.

## 5. Can the partial derivative of x^y be negative?

Yes, the partial derivative of x^y can be negative. This occurs when the exponent y is less than 1. For example, the partial derivative of x^(1/2) would be 1/2x^(-1/2), which is a negative value. This indicates that the function is decreasing in that particular variable.

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