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# Prove formulas for the partial derivatives of xy

Define a function of two variables, with ,

Prove

Proof. First, we write,

Then we compute the derivatives using the chain rule and formulas for the derivatives of the exponential and logarithm,

And,

# Show a given equation satisfies a particular second or partial differential equation

Let

Show that

Proof. First, we compute the first-order partial derivative with respect to ,

Then, we compute the first-order partial derivative with respect to ,

Next, we compute the second-order partial derivatives in the statement we are trying to prove,

Finally, putting these together we have,

Show that

when

1. ;
2. .

1. We compute,

2. We compute,

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

Then,

From this we see that the mixed partials are equal:

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal:

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

Using the chain rule, the first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal:

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal:

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal:

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal:

# Compute partial derivatives of the given function

Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal: