Explanation Given Information: The profit function P = 200 x − 1 2 x 2 . The rate of sales is increasing at the rate of d x d t = 10 units/day . At the instant x = 50 units Formula used: Chain rule: If y = f ( u ) is a differentiable function of u and u = g ( x ) is a differentiable function of x , then y = f ( g ( x ) ) is a differentiable function of x and then d y d x = d y d u ⋅ d u d x . Differentiation formula used: General power rule d d x [ u n ] = n [ u ] n − 1 d u d x . Sum and difference rule d d x ( u ± v ) = d d x ( u ) ± d d x ( v ) . Constant multiple rule d d x ( c u ) = c d d x ( u ) . Calculation: Consider the profit function, P = 200 x − 1 2 x 2 The need is to find d P d t when x = 50 . To calculate the rate of change of the profit d P d t , differentiate both sides with respect to t using the chain rule. d d t ( P ) = d d t ( 200 x − 1 2 x 2 ) Apply sum and difference rule d d x ( u ± v ) = d d x ( u ) ± d d x ( v )

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Chapter 2.8, Problem 3CP

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To calculate: The rate of change of the profit (in dollars) with respect to time (in days), where the profit from selling x units of a product is given by P=200x−12x2. The sales are increasing at a rate of 10 units per day and at the instant x=50 units .

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Chapter 2.8, Problem 2CP

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The rate of change of the profit (in dollars) with respect to time (in days), where the profit from selling x units of a product is given by P = 200 x − 1 2 x 2 . The sales are increasing at a rate of 10 units per day and at the instant x = 50 units .

Solution Summary: The author calculates the rate of change of the profit (in dollars) with respect to time.

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To calculate: The rate of change of the profit (in dollars) with respect to time (in days), where the profit from selling x units of a product is given by P=200x−12x2. The sales are increasing at a rate of 10 units per day and at the instant x=50 units .

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