3. Using the Product Rule, find the derivative these functions: a. f(x)= (2x^4 x

Question

3. Using the Product Rule, find the derivative these functions:

a. f(x)= (2x^4 x^2)(2x^2 + 3x - 6)

b. g(x)= (3x^3 7x)(3X^6 + 2x^4 - 5)

4. Find the derivative of the functions below:

a. f(x) =5x^3 x over
3x^2 + 2

b. f(x) =4x^3 3x^2 + 6x -3 over,
3x^2 + 2x - 7

5. Profit of a company selling videogames follows Profit Function:

P(x) = 50x 0.05x^2 6000

a. Find Rate of change of the company from selling 600 to 700 videogames

b. Find Profit Margin function of the company (P(x))

c. What is the company profit and marginal profit from selling 800 videogames?

d. When will the company make the maximum profit? What is its maximum profit?

with steps, prefer written

Explanation / Answer

Recall that the product rule is

[dfrac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)]

Recall that the quotient rule is

[dfrac{d}{dx}left[dfrac{f(x)}{g(x)} ight] = dfrac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}]

Let (u(x) = 2x^4 - x^2) and (v(x) = 2x^2 + 3x - 6). Then, (u'(x) = 8x^3 - 2x) and (v'(x) = 4x + 3). Thus,

[f'(x) = u'(x)v(x) + u(x)v'(x) = (8x^3 - 2x)(2x^2 + 3x - 6) + (2x^4 - x^2)(4x + 3)]

Let (u(x) = 3x^3 - 7x) and (v(x) = 3x^6 + 2x^4 - 5). Then, (u'(x) = 9x^2 - 7) and (v'(x) = 18x^5 + 8x^3). Thus,

[g'(x) = u'(x)v(x) + u(x)v'(x) = (9x^2 - 7)(3x^6 + 2x^4 - 5) + (3x^3 - 7x)(18x^5 + 8x^3)]

We are given that

[f(x) = dfrac{5x^3 - x}{3x^2 + 2}]

Then,

[f'(x) = dfrac{(3x^2 + 2)(15x^2 - 1) - (5x^3 - x)(6x)}{(3x^2 + 2)^2}]

We are given that

[f(x) = dfrac{4x^3 - 3x^2 + 6x - 3}{3x^2 + 2x - 7}]

Then,

[f'(x) = dfrac{(3x^2 + 2x - 7)(12x^2 - 6x + 6) - (4x^3 - 3x^2 + 6x - 3)(6x + 2)}{(3x^2 + 2x - 7)^2}]

For this problem, the rate of change is determined by

[dfrac{P(700) - P(600)}{700 - 600} = dfrac{P(700) - P(600)}{100}]

Then, we have

[dfrac{50(700) - 0.05(700)^2 - 6000 - (50(600) - 0.05(600)^2 - 6000)}{100} = -15]

The profit margin function is determined by differentiating (P(x)). Thus,

[P'(x) = 50 - 0.1x]

The company profit is (P(800) = 2000), and the marginal profit is (P'(800) = -30)

If the company reaches the maximum profit, then (P'(x) = 0), which implies

[0 = 50 - 0.1x Longrightarrow 50 = 0.1x Longrightarrow x = 500]

So the maximum profit is (P(500) = 6500).

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