Difference between revisions of "022 Sample Final A, Problem 1"
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2) The partial derivative is <math style="verticalalign: 4px">y</math>, since we treat anything not involving <math style="verticalalign: 0px">x</math> as a constant and take the derivative with respect to <math style="verticalalign: 0px">x</math>. In more detail, we have  2) The partial derivative is <math style="verticalalign: 4px">y</math>, since we treat anything not involving <math style="verticalalign: 0px">x</math> as a constant and take the derivative with respect to <math style="verticalalign: 0px">x</math>. In more detail, we have  
−  ::<math style="verticalalign: 0px">\frac{\partial}{\partial x} xy = y\frac{\partial}{\partial x} x = y.</math>  +  ::<math style="verticalalign: 0px">\frac{\partial}{\partial x} xy \,=\, y\frac{\partial}{\partial x} x \,=\, y.</math> 
}  }  
Latest revision as of 08:20, 7 June 2015
Find all first and second partial derivatives of the following function, and demostrate that the mixed second partials are equal for the function
Foundations: 

1) Which derivative rules do you have to use for this problem? 
2) What is the partial derivative of , with respect to ? 
Answers: 
1) You have to use the quotient rule and product rule. The quotient rule says that
so The product rule says This means 
2) The partial derivative is , since we treat anything not involving as a constant and take the derivative with respect to . In more detail, we have

Solution:
Step 1: 

First, we start by finding the first partial derivatives. So we have to take the partial derivative of with respect to , and the partial derivative of with respect to . This gives us the following: 

This gives us the derivative with respect to . To find the derivative with respect to , we do the following: 

Step 2: 

Now we have to find the 4 second derivatives, We have

Also,

Showing the equality of mixed partial derivatives,

Finally,

Final Answer: 

The first partial derivatives are:

The second partial derivatives are:
