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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 62 § 7.2 Partial Derivatives

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 62 Partial Derivatives Computing Partial Derivatives Evaluating Partial Derivatives at a Point Local Approximation of f (x, y) Demand Equations Second Partial Derivative Section Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 62 Partial Derivatives DefinitionExample Partial Derivative of f (x, y) with respect to x: Written, the derivative of f (x, y), where y is treated as a constant and f (x, y) is considered as a function of x alone If, then

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 62 Computing Partial DerivativesEXAMPLE SOLUTION Compute for To compute, we only differentiate factors (or terms) that contain x and we interpret y to be a constant. This is the given function. Use the product rule where f (x) = x 2 and g(x) = e 3x. To compute, we only differentiate factors (or terms) that contain y and we interpret x to be a constant.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 62 Computing Partial Derivatives This is the given function. Differentiate ln y. CONTINUED

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 62 Computing Partial DerivativesEXAMPLE SOLUTION Compute for To compute, we treat every variable other than L as a constant. Therefore This is the given function. Rewrite as an exponent. Bring exponent inside parentheses. Note that K is a constant. Differentiate.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 62 Evaluating Partial Derivatives at a PointEXAMPLE SOLUTION Let Evaluate at (x, y, z) = (2, -1, 3).

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 62 Local Approximation of f ( x, y )

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 62 Local Approximation of f ( x, y )EXAMPLE SOLUTION Let Interpret the result We showed in the last example that This means that if x and z are kept constant and y is allowed to vary near -1, then f (x, y, z) changes at a rate 12 times the change in y (but in a negative direction). That is, if y increases by one small unit, then f (x, y, z) decreases by approximately 12 units. If y increases by h units (where h is small), then f (x, y, z) decreases by approximately 12h. That is,

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 62 Demand EquationsEXAMPLE SOLUTION The demand for a certain gas-guzzling car is given by f (p 1, p 2 ), where p 1 is the price of the car and p 2 is the price of gasoline. Explain why is the rate at which demand for the car changes as the price of the car changes. This partial derivative is always less than zero since, as the price of the car increases, the demand for the car will decrease (and visa versa). is the rate at which demand for the car changes as the price of gasoline changes. This partial derivative is always less than zero since, as the price of gasoline increases, the demand for the car will decrease (and visa versa).

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 62 Second Partial DerivativeEXAMPLE SOLUTION Let. Find We first note that This means that to compute, we must take the partial derivative of with respect to x.

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