Formulas for Derivatives: Business Calculus Section 1.3 Study Notes

Formulas for Derivatives

Introduction

This section introduces the fundamental rules for finding derivatives, which are essential for analyzing rates of change in business, economics, and related fields. These rules allow us to compute derivatives efficiently without reverting to the limit definition each time.

Derivative Rules: Building Blocks

The following are the basic rules for differentiating common types of functions. These rules form the foundation for more advanced calculus techniques.

Rule Name	Formula
Derivative of a Constant (k) Rule	$\frac{d}{dx}(k) = 0$
Constant Multiple Rule	$\frac{d}{dx}[k \cdot f(x)] = k \cdot f'(x)$
Sum or Difference Rule	$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
Power Rule	$\frac{d}{dx}[x^n] = n x^{n-1}$
Identity Function Rule	$\frac{d}{dx}(x) = 1$

Examples and Applications

• Derivative of a Constant: The derivative of any constant function is zero. Example: $y = 4$ → $y' = 0$

• Derivative of a Linear Function: The derivative of $y = mx + b$ is its slope $m$. Example: $y = 2x + 3$ → $y' = 2$

• Power Rule: For $y = x^2$, $y' = 2x$. For $y = x^n$, $y' = n x^{n-1}$.

• Constant Multiple Rule: Constants are carried along during differentiation. Example: $y = 4x^2$ → $y' = 4 \cdot 2x = 8x$

• Sum/Difference Rule: Differentiate each term separately. Example: $y = 5x^2 + 0.5x^2 - 10x + 157$ $y' = 5 \cdot 2x + 0.5 \cdot 2x - 10 + 0 = 10x + x - 10$

• Negative and Fractional Powers: The power rule applies to all real exponents. Example: $y = x^{-2}$ → $y' = -2x^{-3}$

• Roots as Exponents: $\sqrt[n]{x^m} = x^{m/n}$ Example: $y = \sqrt{x} = x^{1/2}$ → $y' = \frac{1}{2}x^{-1/2}$

Business Applications: Marginal Cost, Revenue, and Profit

Derivatives are used to analyze how cost, revenue, and profit change as the number of items produced or sold changes. The concept of a marginal function is central in economics and business calculus.

• Marginal Cost (MC): The derivative of the cost function $C(x)$. $MC(x) = C'(x) = \frac{dC}{dx}$

• Marginal Revenue (MR): The derivative of the revenue function $R(x)$. $MR(x) = R'(x) = \frac{dR}{dx}$

• Marginal Profit (MP): The derivative of the profit function $P(x) = R(x) - C(x)$. $MP(x) = P'(x) = R'(x) - C'(x) = MR(x) - MC(x)$

These marginal functions approximate the change in cost, revenue, or profit when producing or selling one additional unit.

Example: Marginal Cost Calculation

• Given $C(x) = \sqrt{x}$ (cost in hundreds of dollars), find $MC(100)$.

• $C'(x) = \frac{1}{2}x^{-1/2}$, so $MC(100) = \frac{1}{2}(100)^{-1/2} = \frac{1}{2 \cdot 10} = 0.05$ (hundred dollars, or $5).

Finding Tangent Lines

The equation of the tangent line to the graph of $f(x)$ at $x = a$ is:

• Point: $(a, f(a))$

• Slope: $f'(a)$

• Equation: $y - f(a) = f'(a)(x - a)$

Example: For $f(x) = 10 - x^2$ at $x = 2$, $f'(x) = -2x$, so $f'(2) = -4$. The tangent line is $y - 6 = -4(x - 2)$.

Practice Exercises

Students are encouraged to practice by finding derivatives of various polynomial, root, and rational functions, as well as applying marginal analysis to cost, revenue, and profit functions.

Summary Table: Derivative Rules

Function	Derivative
$k$	$0$
$x$	$1$
$x^n$	$n x^{n-1}$
$k f(x)$	$k f'(x)$
$f(x) \pm g(x)$	$f'(x) \pm g'(x)$

Key Takeaways

• Mastery of derivative rules is essential for efficient problem-solving in calculus.

• Marginal analysis is a powerful tool for business applications, allowing for the estimation of changes in cost, revenue, and profit.

• Practice with a variety of functions and applications is necessary for proficiency.