Question 1

A pharmaceutical company models the concentration of a drug in the bloodstream over time as $$y = 	ext{sin}(6t)$$, where $$t is in $$hours. What is the rate of change of the drug concentration at $$t = 2$$ hours?

Accepted Answer

$$y' = 6 	ext{cos}(12)$$

Question 2

A company’s profit over time is modeled by $$y = 	ext{cos}(2t)$$, where $$t is in $$years. What is the derivative of the profit function with respect to time?

Accepted Answer

$$y' = -2 	ext{sin}(2t)$$

Question 3

A marketing analyst models seasonal demand as $$y = 	ext{sin}^2(x). $$What is the derivative of $$y$$ with respect to $$x$$?

Accepted Answer

$$y' = 2 	ext{sin}(x) 	ext{cos}(x)$$

Question 4

A company’s cost function is $$y = 	ext{tan}(2x)$$, where $$x is $$the number of units produced. What is the marginal cost (the derivative) with respect to $$x$$?

Accepted Answer

$$y' = 2 	ext{sec}^2(2x)$$

Question 5

A business models its revenue as $$y = e$$^{x}$$ 	ext{cos}(x). $$What is the derivative of the revenue function?

Accepted Answer

$$y' = e$$^{x}$$ (	ext{cos}(x) - 	ext{sin}(x))$$

Question 6

A company’s advertising effectiveness is modeled by $$y = 	ext{sec}(6x^2$)$$, where $$x is $$the number of weeks. What is the derivative of $$y$$ with respect to $$x$$?

Accepted Answer

$$y' = 12x 	ext{sec}(6x^2$) 	ext{tan}(6x^2$)$$

Question 7

A company’s cost function is $$y = x^2$ 	ext{cos}(x). $$What is the derivative of the cost function?

Accepted Answer

$$y' = 2x 	ext{cos}(x) - x^2$ 	ext{sin}(x)$$

Question 8

A company’s profit function is $$y = 	ext{sin}(x) + 	ext{cos}(x). $$What is the derivative of the profit function?

Accepted Answer

$$y' = 	ext{cos}(x) - 	ext{sin}(x)$$

Question 9

A company’s cost function is $$y = rac{	ext{sin}(x)}{x}. $$What is the derivative of the cost function?

Accepted Answer

$$y' = rac{x 	ext{cos}(x) - 	ext{sin}(x)}{x^2$}$$

Question 10

A company’s revenue function is $$y = e$$^{x}$$ 	ext{sin}(x). $$What is the derivative of the revenue function?

Accepted Answer

$$y' = e$$^{x}$$ (	ext{sin}(x) + 	ext{cos}(x))$$

Question 11

A company’s cost function is $$y = 	ext{tan}^2(x). $$What is the derivative of the cost function?

Accepted Answer

$$y' = 2 	ext{tan}(x) 	ext{sec}^2(x)$$

Question 12

A company’s profit function is $$y = 	ext{cos}(x^2$ + 5x - 1). $$What is the derivative of the profit function?

Accepted Answer

$$y' = -	ext{sin}(x^2$ + 5x - 1) (2x + 5)$$

Question 13

A company’s cost function is $$y = 	ext{sin}(6x^2$ + 3x + 2). $$What is the derivative of the cost function?

Accepted Answer

$$y' = 	ext{cos}(6x^2$ + 3x + 2) (12x + 3)$$

Question 14

A company’s profit function is $$y = rac{2 	ext{cos}(x) + 4 	ext{sin}(x)}{	ext{cos}(4x)}. $$What is the derivative of the profit function?

Accepted Answer

$$y' = rac{-2 	ext{sin}(x) + 4 	ext{cos}(x)}{	ext{cos}(4x)} - rac{2 	ext{cos}(x) + 4 	ext{sin}(x)}{	ext{cos}^2(4x)} 	imes (-4 	ext{sin}(4x))$$

Question 15

A company’s cost function is $$y = 	ext{sec}^2(x). $$What is the derivative of the cost function?

Accepted Answer

$$y' = 2 	ext{sec}(x) 	ext{sec}(x) 	ext{tan}(x)$$

Question 16

A company’s profit function is $$y = 	ext{sin}(x) 	ext{cos}(x). $$What is the derivative of the profit function?

Accepted Answer

$$y' = 	ext{cos}^2(x) - 	ext{sin}^2(x)$$

Question 17

A company’s cost function is $$y = 	ext{sin}(x^2$). $$What is the derivative of the cost function?

Accepted Answer

$$y' = 2x 	ext{cos}(x^2$)$$

Question 18

A company’s profit function is $$y = 	ext{cos}(x) - 	ext{sin}(x). $$What is the derivative of the profit function?

Accepted Answer

$$y' = -	ext{sin}(x) - 	ext{cos}(x)$$

Derivatives of Trigonometric Functions: Business Calculus Study Notes

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