QuestionFind the marginal profit $P^{\prime}(x)$ for $P(x)=3 \cdot x-8 \cdot \sqrt{x}$ and calculate $\frac{P^{\prime}(x)}{x}$ .

Studdy Solution

STEP 1

Assumptions1. The profit function is given by $(x)=3 \cdot x-8 \cdot \sqrt{x}$ . We need to find the marginal profit, which is the derivative of the profit function, $^{\prime}(x)$ 3. We also need to find the marginal profit per item produced, which is the marginal profit divided by the number of items produced, $\frac{^{\prime}(x)}{x}$

STEP 2

First, we need to find the derivative of the profit function, $(x)$ . This is done using the power rule for differentiation, which states that the derivative of $x^n$ is $n \cdot x^{n-1}$ .

STEP 3

Rewrite the square root in the profit function as a power of1/2.
$(x)=3 \cdot x-8 \cdot x^{1/2}$

STEP 4

Now, differentiate the profit function with respect to $x$ .
$^{\prime}(x)=\frac{d}{dx}(3 \cdot x)-\frac{d}{dx}(8 \cdot x^{1/2})$

STEP 5

Differentiate each term separately.
$^{\prime}(x)=3-\frac{8}{2} \cdot x^{-1/2}$

STEP 6

implify the derivative.
$^{\prime}(x)=3-4 \cdot x^{-1/2}$

STEP 7

Now that we have the marginal profit, $^{\prime}(x)$ , we can find the marginal profit per item produced, $\frac{^{\prime}(x)}{x}$ .

STEP 8

Divide the marginal profit by the number of items produced.
$\frac{^{\prime}(x)}{x}=\frac{3-4 \cdot x^{-1/2}}{x}$

STEP 9

implify the expression by multiplying the numerator and denominator by $x^{/2}$ .
$\frac{^{\prime}(x)}{x}=3 \cdot x^{-/2}-4 \cdot x^{-3/2}$ So, the marginal profit is $^{\prime}(x)=3-4 \cdot x^{-/2}$ and the marginal profit per item produced is $\frac{^{\prime}(x)}{x}=3 \cdot x^{-/2}-4 \cdot x^{-3/2}$ .

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