Calculus 220 section 7 2 Partial Derivatives notes by Tim Pilachowski When working with functions of more than one variable the question in calculus becomes how can we evaluate the rate of change The answer is called a partial derivative Given a function f x y z the partial f derivative of f with respect to x is found by treating all variables other than x as constants The partial x f f derivatives and have analogous definitions y z Example A Given the function f x y 2x 2y find f f and Answers 2 2 x y Example B Given the function f x y 9 x 2 y 2 find f f and Answers 2x 2y x y A geometric interpretation of partial derivative is pictured below In each figure f x y is the curved surface In the figure on the left with y treated as a constant the tangent line goes the same general direction as the x axis f and is the slope of that tangent at point P With x treated as a constant the tangent line goes the same x f general direction as the y axis and is the slope of that tangent at point P y x f f f ln z find and x y z y 1 x x Answers 2 xyz yze xy ln z x 2 z xze xy ln z x 2 y e xy 2 y yz y Example C Given the function f x y z x 2 yz ze xy Example D For f x y x 2 3 xy y 7 determine f 5 8 and f 5 8 Answers 34 16 dx dy Example E The revenue generated by x model A speakers and y model B speakers is given by R x y 100 x 150 y 0 03x 2 0 02 y 2 dollars Determine the rate at which revenue will change with respect to the change in the number of model A speakers sold when 50 model A speakers and 40 model B speakers have been sold Answer 97 per model A speaker Example F Example C from Lecture 7 1 revisited For a particular manufacturing plant the number of units produced is given by the Cobb Douglas function f x y 20 x 0 4 y 0 6 where x the number of units of labor f 1200 2000 and f 1200 2000 and y the number of units of capital Find and interpret x y Answers 11 units 10 units f is called the marginal productivity of labor At the current production level using x 1200 units of labor and 2000 units of capital if labor is increased by one unit the level of production will increase by approximately 11 units This partial derivative f is called the marginal productivity of capital At the current production level using y 1200 units of labor and 2000 units of capital if capital is increased by one unit the level of production will increase by approximately 10 units This partial derivative 2 f 2 f x 2 y 2 x y y x Just like regular derivatives higher order partial derivatives can be found The first two are called respectively the second partial derivative with respect to x and the second partial derivative with respect to y The second two are often called mixed partial derivatives In this case as with most functions the two mixed partials are equal You should use this fact to check your answers Answers yex 6xy 3y2 ex 3y2 ex Example G For f x y xy 3 ye x determine 2 f 2 f 2 f xy Example H For g x y ln xy e determine y x Just like regular derivatives you need to know and be able to use the product rule quotient rule and chain rule Answer xyexy exy