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Calculus_MA_224_Chapter_7_Section_2

# Calculus_MA_224_Chapter_7_Section_2 - f = f Chain Rule for...

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Calculus MA 224 Chapter 7 Section 2: Partial Derivatives Partial Differentiation : Differentiating the function with respect to the particular variable in question while keeping the other variable fixed. The resulting derivative is said to be a Partial Derivative of the function Partial Derivatives : The partial derivative of f with respect to x is denoted by f (x,y) With respect to y: f(x,y) The other variable is treated as a constant Ex. 1 Find the partial derivatives f and f if f(x,y) = Simplify: f(x,y)= f(x,y)= f(x,y)= 0 + 2x(2y) + Second-Order Partial Derivatives : If z= f(x,y), the partial derivative of f with respect to x is The partial derivative of f with respect to y is The partial derivative of f with respect to x is The partial derivative of f with respect to y is = Ex. 2 Compute the four second-order partial derivatives of the function f(x,y) = xy + 5xy + 2x + 1

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Solution: f = y + 5y + 2 so f= 0 and f= 3y + 10y f = 3xy + 10xy so f = 6xy +10x and f = 3y + 10y
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Unformatted text preview: f = f Chain Rule for Partial Derivatives : Suppose z is a function of x and y, each of which is a function of t. Then z can be regarded as a function of t and Incremental Approximation Formula For Functions of Two Variables : Suppose z is a function of x and y. If x denotes a small change in x and y a small change in y, the corresponding change in z is Ex. 3 At a certain factory, the daily output is Q= 60KL units, where K denotes the capital investment measured in units of \$1000 and L the size of the labor force measured in worker-hours. The current capital investment is \$900,000, and 1,000 worker-hours of labor are used each day. Estimate the change in output that will result if capital investment is increased by \$1,000 and labor in increased by 2 worker-hours. Solution: Apply the approximation formula with K = 900, L = 1,000, K = 1, and L = 2 to get, =30KLK+20KLL =30(1/30)(10)(1)+20(30)(1/100)(2) =22 units Output will increase by 22 units...
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Calculus_MA_224_Chapter_7_Section_2 - f = f Chain Rule for...

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