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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, # Prove two formulas with partial derivatives

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: 