# The profit from the expenditure of x thousand dollars on advertising is given by P(x) = 1040 +...

## Question:

The profit from the expenditure of {eq}x {/eq} thousand dollars on advertising is given by {eq}P(x) = 1040 + 25x - 3x^2 {/eq}.

Find the marginal profit when the expenditure is {eq}x = 9 {/eq}.

## Marginal Profit:

Suppose we are given the profit function {eq}P(x) {/eq}.

If we need to find the marginal profit, then we should differentiate {eq}P(x) {/eq}.

This is because the marginal profit pertains to the rate of change of the profit function and remember that we differentiate a function to attain its rate of change.

## Answer and Explanation:

To attain the marginal profit function, we differentiate the given profit function:

{eq}\begin{align*} P(x) & = 1040 + 25x - 3x^2\\ P'(x) & = 25 - 6x \ \ \ \left[\mathrm{ Differentiation \ Power \ Rule: \ }D_x x^n = n x^{n-1}\right] \\ \end{align*} {/eq}

Determine the marginal profit when the expenditure is {eq}x=9 {/eq} by plugging in {eq}x=9 {/eq} into the marginal profit function:

{eq}\begin{align*} P'(x) & = 25 - 6x\\ P'(9) & = 25 - 6(9)\\ \implies P'(9) & = -29\\ \end{align*} {/eq}

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