Chapter 1: Differentiation – Study Guide for Business Calculus

Chapter 1: Differentiation

Limits and Continuity

Understanding limits and continuity is foundational for differentiation. Limits describe the behavior of a function as the input approaches a particular value, while continuity ensures a function behaves predictably without breaks or jumps.

• Finding Limits Using Graphs: To estimate $\lim_{x \to a} f(x)$ from a graph, observe the value that $f(x)$ approaches as $x$ gets close to $a$ from both sides.

• Limit Properties: Limits can be calculated using properties such as linearity, product, and quotient rules. For example:

  • $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

  • $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$

• Continuity at a Point: A function $f$ is continuous at $x = a$ if:

  • $f(a)$ exists

  • $\lim_{x \to a} f(x)$ exists

  • $\lim_{x \to a} f(x) = f(a)$

• Using Graphs to Determine Continuity: Check for holes, jumps, or vertical asymptotes at the point or interval in question.

Difference Quotients and Average Rate of Change

The difference quotient is a formula that gives the average rate of change of a function over an interval. It is foundational for understanding derivatives.

• Difference Quotient: $\displaystyle \frac{f(x + h) - f(x)}{h}$

• Application: Used to compute the average rate of change of $f$ over $[x, x+h]$.

• Example: For $f(x) = x^2$, the difference quotient is $\displaystyle \frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$.

Derivatives and Their Interpretation

The derivative measures the instantaneous rate of change of a function. It is defined as the limit of the difference quotient as $h$ approaches zero.

• Definition of the Derivative:

  • $f'(x) = \displaystyle \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Graphical Interpretation: The derivative at $x$ is the slope of the tangent line to the graph of $f$ at $x$.

• Differentiability: A function is differentiable at $x$ if the derivative exists at $x$. If the graph has a sharp corner, cusp, or vertical tangent, the function may not be differentiable there.

Basic Differentiation Rules

Several rules simplify the process of finding derivatives for common functions and combinations of functions.

• Power Rule: $\displaystyle \frac{d}{dx} [x^n] = n x^{n-1}$

• Constant Multiple Rule: $\displaystyle \frac{d}{dx} [c \cdot f(x)] = c \cdot f'(x)$

• Sum and Difference Rules: $\displaystyle \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$

• Example: $\displaystyle \frac{d}{dx} [3x^2 + 5x] = 3 \cdot 2x + 5 = 6x + 5$

Evaluating Derivatives and Tangent Lines

Once the derivative is found, it can be evaluated at specific points to find slopes or to write equations of tangent lines.

• Value of Derivative at a Point: Substitute the $x$-value into $f'(x)$.

• Equation of the Tangent Line: The tangent line at $x = a$ is $y = f(a) + f'(a)(x - a)$.

• Example: For $f(x) = x^2$ at $x = 1$, $f'(1) = 2$, so the tangent line is $y = 1 + 2(x - 1)$.

Leibniz Notation and Applications

Leibniz notation expresses derivatives as $\frac{dy}{dx}$, which is especially useful in applications and when dealing with related rates or implicit differentiation.

• Notation: $\frac{dy}{dx}$ or $\frac{df}{dx}$

• Application: Used to express rates of change in applied problems, such as velocity ($\frac{ds}{dt}$) or marginal cost ($\frac{dC}{dx}$).

Product and Quotient Rules

When differentiating products or quotients of functions, use the following rules:

• Product Rule: $\displaystyle \frac{d}{dx} [f(x) \cdot g(x)] = f'(x) \cdot g(x) + g'(x) \cdot f(x)$

• Quotient Rule: $\displaystyle \frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}$

• Example (Product Rule): If $f(x) = x^2$ and $g(x) = \sin x$, then $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$.

• Example (Quotient Rule): If $f(x) = x^2$ and $g(x) = x+1$, then $\frac{d}{dx} \left[ \frac{x^2}{x+1} \right] = \frac{(x+1)2x - x^2 \cdot 1}{(x+1)^2}$.

Chain Rule and Extended Power Rule

The chain rule is used to differentiate composite functions, while the extended power rule generalizes the power rule to functions raised to a power.

• Chain Rule: $\displaystyle \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$

• Extended Power Rule: $\displaystyle \frac{d}{dx} [g(x)]^k = k [g(x)]^{k-1} \cdot g'(x)$

• Example (Chain Rule): $\frac{d}{dx} [(3x+2)^5] = 5(3x+2)^4 \cdot 3 = 15(3x+2)^4$

Higher-Order Derivatives

Higher-order derivatives are derivatives of derivatives, representing rates of change of rates of change (e.g., acceleration is the second derivative of position).

• Notation: $f''(x)$ for the second derivative, $f'''(x)$ for the third, or $f^{(n)}(x)$ for the $n$th derivative.

• Application: In business, the second derivative can indicate concavity or the rate at which marginal cost or revenue is changing.

• Example: If $f(x) = x^3$, then $f'(x) = 3x^2$ and $f''(x) = 6x$.

Summary Table: Differentiation Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx} [x^n] = n x^{n-1}$	$\frac{d}{dx} [x^3] = 3x^2$
Constant Multiple Rule	$\frac{d}{dx} [c \cdot f(x)] = c \cdot f'(x)$	$\frac{d}{dx} [5x^2] = 5 \cdot 2x = 10x$
Sum/Difference Rule	$\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$	$\frac{d}{dx} [x^2 + x] = 2x + 1$
Product Rule	$\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + g'(x) f(x)$	$\frac{d}{dx} [x^2 \sin x] = 2x \sin x + x^2 \cos x$
Quotient Rule	$\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}$	$\frac{d}{dx} \left[ \frac{x^2}{x+1} \right] = \frac{(x+1)2x - x^2 \cdot 1}{(x+1)^2}$
Chain Rule	$\frac{d}{dx} [f(g(x))] = f'(g(x)) g'(x)$	$\frac{d}{dx} [(3x+2)^5] = 15(3x+2)^4$

Applications in Business Calculus

Differentiation is widely used in business for analyzing marginal cost, marginal revenue, and optimization problems. Understanding how to compute and interpret derivatives is essential for making informed business decisions.

• Marginal Cost: The derivative of the cost function, representing the cost of producing one more unit.

• Marginal Revenue: The derivative of the revenue function, representing the additional revenue from selling one more unit.

• Optimization: Derivatives help find maximum or minimum values, such as maximizing profit or minimizing cost.