1.7

Derivatives in Business Calculus 1.7

Basic Derivative Computation

Derivatives are fundamental tools in calculus, used to measure the rate of change of a function with respect to its variable. In business contexts, derivatives help analyze how quantities such as cost, revenue, and profit change as production levels vary.

• Notation: The derivative of a function $f(x)$ with respect to $x$ is written as $\frac{d}{dx}[f(x)]$ or $f'(x)$.

• Example 1: Compute $\frac{dP}{dt}$ given $P = \frac{4}{\sqrt{(4t^5 - Q)^3}}$.

  • Apply the chain rule and power rule to differentiate with respect to $t$.

• Example 2: Compute $\frac{d}{dr}(7r^4 - \frac{8}{r})$.

  • Use the power rule and the derivative of $r^{-1}$.

• Example 3: Compute $\frac{d}{dr}(r^7)$.

  • Apply the power rule: $\frac{d}{dr}(r^n) = n r^{n-1}$.

Second Derivative

The second derivative of a function measures the rate of change of the rate of change, providing information about concavity and acceleration in business models.

• Notation for $f(x)$:

  • $\frac{d}{dx}[f(x)] = f'(x)$

  • $\frac{d}{dx}[f'(x)] = f''(x)$

• Evaluating at a specific value:

  • For $x = 3$, write $f'(3)$ and $f''(3)$.

• Notation for $y$:

  • $\frac{dy}{dx} = y'$

  • $\frac{d}{dx}[y'] = \frac{d^2y}{dx^2} = y''$

• Evaluating at $x = 3$:

  • $\left.\frac{dy}{dx}\right|_{x=3}$ and $\left.\frac{d^2y}{dx^2}\right|_{x=3}$

• Example: For $y = 7t^6 - 7.8t^4 + t - 499.2$, compute $\left.\frac{d^2y}{dt^2}\right|_{t=3}$.

Marginal Analysis in Business Calculus

Marginal Cost, Revenue, and Profit

In business calculus, functions such as cost, revenue, and profit are often expressed in terms of the number of items produced, $x$. The derivative of these functions at a specific production level estimates the change in cost, revenue, or profit if one more item is produced.

• Marginal Cost: If $C(x)$ is a cost function, the marginal cost function is $C'(x)$.

• Interpretation: The marginal cost of producing $a$ units is approximately equal to the additional cost incurred when production increases from $a$ to $a+1$ units.

• Formula:

  • $C'(a) \approx C(a+1) - C(a)$

Applications and Examples

Marginal analysis is used to make decisions about production levels, pricing, and resource allocation.

• Example: Given $R(45) = 17.5$, $R'(45) = -2.75$, $C(45) = 15.4$, and $C'(45) = 3.12$:

• a. Find the profit at a production level of 45 items:

  • Profit function: $P(x) = R(x) - C(x)$

  • At $x = 45$: $P(45) = R(45) - C(45) = 17.5 - 15.4 = 2.1$

• b. Predict the profit at a production level of 46 items:

  • Approximate change in profit: $P'(45) = R'(45) - C'(45) = -2.75 - 3.12 = -5.87$

  • Estimated profit at 46 items: $P(46) \approx P(45) + P'(45) = 2.1 + (-5.87) = -3.77$

Marginal Revenue Interpretation

Marginal revenue at a specific production level indicates the approximate increase in revenue from selling one additional unit.

• Example: What is the marginal revenue at a production level of 100 items?

• Interpretation: The value of $R'(100)$ tells us how much revenue will increase (or decrease) if production is increased from 100 to 101 items.

Summary Table: Marginal Functions in Business Calculus

Function	Marginal Function	Interpretation
Cost $C(x)$	Marginal Cost $C'(x)$	Approximate additional cost for one more unit
Revenue $R(x)$	Marginal Revenue $R'(x)$	Approximate additional revenue for one more unit
Profit $P(x)$	Marginal Profit $P'(x)$	Approximate additional profit for one more unit

Key Formulas

• Derivative of a power function: $\frac{d}{dx}(x^n) = n x^{n-1}$

• Marginal cost approximation: $C'(a) \approx C(a+1) - C(a)$

• Profit function: $P(x) = R(x) - C(x)$

• Marginal profit: $P'(x) = R'(x) - C'(x)$

Additional info:

• Marginal analysis is a cornerstone of decision-making in economics and business, allowing firms to optimize production and pricing strategies.

• Second derivatives can be used to analyze the concavity of cost, revenue, and profit functions, which is important for identifying maximum and minimum values (optimization).