Chapter 15

1. Chapter 15 Section 15.6 Limits and Continuity; Equality of Mixed Partials

2. Second Derivatives For a function of several variables there is more than one first derivative. I fact if there are 2 first derivatives ( and ) and 4 second derivatives (really 3 different ones). Second derivatives of Take derivative of z with respect to x twice. Take derivative of z first with respect to y then x. Take derivative of z first with respect to x then y. Take derivative of z with respect to y twice. Example Find all first and second derivatives of the function Notice that the mixed partial derivatives are equal. Is this a coincidence? NO!

3. Equality of Mixed Partials At all points of continuity of the second derivative the mixed partials will be equal. This says it does not matter what order you take the higher derivatives in. For Higher Order Derivatives Derivatives higher than 2 can also be taken it is a matter of how many times this is done with respect to each variable. If you take the derivative h times with respect to x and k times with respect to y this is one of the derivatives To find Take derivative of z with respect to x variable h times and then take it with respect to y variable k times. Example For the function find . This means take the derivative of this function 4 times, 3 of them treating x as the variable and one of them treating y as the variable.