Worksheet 2.2: Partial Derivatives

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1 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives From the Toolbox (what you need from previous classes) Be familiar with the definition of a derivative as the slope of a tangent line (the limit of the slopes of secant lines) from Calc I. Be able to find the slope of a line. Be able to compute derivatives of single-variable functions y = f (x) using techniques learned in Calc I. Partial derivatives of multivariate functions are computed using methods from Calc I, treating one of the independent ( input ) variables as the variable of differentiation, and treating all others as though they were constants Geometrically, partial derivatives are slopes of tangent lines to curves in the graph of the function (specifically, vertical traces). Higher-order partial derivatives are derivatives of derivatives. In this worksheet, you will: Explore the geometric meaning of a partial derivative, see how that is similar to the geometric meaning of derivatives from Calc I, and use it to justify using computational techniques from Calc I to compute partial derivatives. Compute first-, second-, and higher-order partial derivatives.

2 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives 1 Model 1: Relationship of Partial Derivatives to Derivatives of Single-Variable Functions (Calc I - Type Derivatives) Diagram 2A Diagram 2B Shown here are the: Graph of the function f (x, y ) = x 2 + y 2. Vertical trace for x = 1. This is a curve in the graph. Tangent line to the vertical trace at P = (1, 1, 2). Shown here is the plane x = 1. This is the plane that intersects the graph of the function f (x, y) = x 2 +y 2, producing the vertical trace pictured in Diagram 2A. Critical Thinking Questions In this section, you will explore why you can use the techniques from Calc I to compute derivatives of functions of two (or more) variables. (Q1) Recall from Worksheet 2.1 (Model 2): a vertical trace is a curve in the graph of a function that is generated by holding one of the variables constant. For the vertical trace shown in Diagram 2A, the variable x / y / z is being held constant, at the value x =. (Q2) Since x = 1 along this trace, you can write an equation for the vertical trace as a function of only one variable y, by replacing x with the value 1: F (y) = f (1, y) = ( ) 2 + y 2 = + y 2 (Q3) Sketch the graph of the function F (y) = 1 + y 2 from (Q2) on Diagram 2B.

3 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives 2 (Q4) Compute the derivative F (y) for the function F (y) = 1 + y 2 from (Q2). function of a single variable, you compute the derivative using the from Calc I. Since this is a rule Then, evaluate the derivative at y = 1, which is the y-coordinate of the point P in Model 1. F (y) = F (1) = (Q5) The derivative F (1) = 2 from (Q4) is the slope of the line tangent to the graph of F (y) at the point P. You have already graphed the function F (y) on Diagram 2B; now, add this tangent line to your sketch at the point P. Compare your sketch in Diagram 2B to the original Diagram 2A. Your sketch of the graph of F (y) = 1 + y 2 and its tangent line at P in Diagram 2B is the vertical trace and tangent line from Diagram 2A. ( Q6) Below is a contour map for a function f (x, y). Use it to determine the signs of the partial derivatives of f at the point P = (a, b). f x (a, b) is: positive / negative / zero f y (a, b) is: positive / negative / zero

4 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives 3 Model 2: Second Derivatives & Higher Order Derivatives Leibnitz Notation: The notation 2 f means: First, compute the partial derivative with respect to : Then, compute the partial derivative with respect to : So: 2 f = [ ] f f [ ] f Subscript Notation: The notation f means: First, compute the partial derivative with respect to : Then, compute the partial derivative with respect to : ) So: f = (f f (f ) Critical Thinking Questions In this section, you will compute higher-order derivatives of multivariate functions. (Q7) Computing a second derivative means computing two derivatives. first, you compute the derivative of a function, then you compute the derivative of the first derivative. Refer to Model 2. The notation 2 f means: first compute the partial derivative of f with x 2 respect to the variable, then compute the derivative of the first derivative with respect to the variable. (Q8) Refer to Model 2. The notation f yy means: first compute the partial derivative of f with respect to the variable, then compute the derivative of the first derivative with respect to the variable. 2 f (Q9) Refer to Model 2. The notation means: first compute the partial derivative of f with x y respect to the variable, then compute the derivative of the first derivative with respect to the variable. (Q10) Refer to Model 2. The notation f xy means: first compute the partial derivative of f with respect to the variable, then compute the derivative of the first derivative with respect to the variable.

5 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives 4 (Q11) For a single-variable function f (t), there is only one second derivative: f (t) (also denoted d 2 f /dt 2 ). For a function f (x, y), there are four second partial derivatives. Referring to Model 2, write these out symbolically, using both Leibnitz and subscript notation. (Q12) Let f (x, y) = 10x 2 y 3 5x 4 + 7y. Compute 2 f x 2, 2 f y 2, 2 f y x, and 2 f x y. (Q13) Let g(x, y) = 5xy + x 2 e 2y First, compute g xy (x, y) (x-derivative first, then y- derivative). Then, compute g yx (x, y) (y-derivative first, then x-derivative). (Q14) The second partial derivatives f xy and f yx are called mixed partial derivatives. What do you notice about the mixed partial derivatives you computed in (Q12,13)?

6 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives 5 Clairaut s Theorem: If f (x, y) is defined on an open disk D, and f xy and f yx are continuous on D, then for all points (a, b) in D: f xy (a, b) = f yx (a, b) Informally, mixed partial derivatives commute. This holds for functions of more than two variables as well. (Q15) Let h(x, y) = tan 2 (e y ) + ln ( y 2 /(1 + y 2 ) ) + x 2 y. If you want to compute the second mixed partial derivative, the smart way would be to compute the first derivative with respect to x / y. (Q16) Compute the derivative from (Q15). (Q17) Computing higher-order partial derivatives follows the same pattern as second-order partial derivatives. For example, the third partial derivative of f first with respect to x, then with 3 f respect to y, then again with respect to x is f xyx or x y x. Let f (x, y) = x 3 y x. Compute f xyx, f yyx, f yyy.

7 Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives 6 (Q18) Computing partial derivatives of functions of more than two variables follows the same pattern as partial derivatives of functions of two variables. Let g(x, y, z) = ln(x 2 + y 2 + z 2 ) (a) Compute g x (x, y, z), g y (x, y, z), and g z (x, y, z). (b) Compute g xx (x, y, z), g xy (x, y, z), and g xz (x, y, z). ( Q19) In (Q11), you saw that there are four second partial derivatives for a function f (x, y). Assuming Clairaut s Theorem holds (mixed partial derivatives are the same), how many of these are distinct functions? ( Q20) How many second derivatives are there in total for a function of 3 variables, f (x, y, z)? Assuming Clairaut s Theorem holds, how many of these are distinct functions? ( Q21) How many second derivatives are there in total for a function of n variables, f (x 1, x 2,..., x n )? Assuming Clairaut s Theorem holds, how many of these are distinct functions?