Differentiation and Applications in Calculus for Economics

Differentiation and Its Applications

Limits

Limits are fundamental in calculus, providing a precise way to describe the behavior of functions as their input values approach certain points. They are essential for defining derivatives and understanding continuity.

• Definition: The limit of a function f(x) as x approaches a is denoted by $\lim_{x \to a} f(x)$.

• Application: Used to analyze economic and financial metrics as they approach thresholds.

• Direct Substitution: For most functions, the limit can be found by substituting the value directly.

• Example: $\lim_{x \to 2} (x^2 + 1) = 2^2 + 1 = 5$

• Non-Existence: Some limits do not exist, such as $\lim_{x \to 2} \frac{1}{x-2}$, which approaches infinity as x approaches 2.

Secant and Tangent Lines

The secant line connects two points on a curve, while the tangent line touches the curve at a single point. As the points get closer, the secant line approaches the tangent line.

• Slope of Secant: $\frac{\Delta y}{\Delta x} = \frac{f(x+h) - f(x)}{h}$

• Limit Definition of Derivative: $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Derivative: Definition and Notation

The derivative measures the rate of change of a function with respect to its variable. It is foundational in calculus and economics for analyzing marginal changes.

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

• Notations: $y'$, $f'(x)$, $\frac{dy}{dx}$, $\frac{df(x)}{dx}$, $\frac{d}{dx}f(x)$

• Example: For $f(x) = x^2 + x + 2$, $f'(x) = 2x + 1$

Derivatives of Common Functions

Several rules allow for efficient computation of derivatives for common functions.

Function	Derivative
$y = x^n$	$y' = nx^{n-1}$
$y = e^x$	$y' = e^x$
$y = e^{g(x)}$	$y' = g'(x)e^{g(x)}$
$y = \ln(x)$	$y' = \frac{1}{x}$
$y = \ln(g(x))$	$y' = \frac{g'(x)}{g(x)}$

• Example: $y = x^5 \Rightarrow y' = 5x^4$

• Example: $y = \frac{1}{x^3} = x^{-3} \Rightarrow y' = -3x^{-4}$

• Example: $y = \sqrt{x} = x^{1/2} \Rightarrow y' = \frac{1}{2\sqrt{x}}$

Rules of Differentiation

These rules simplify the process of finding derivatives for more complex functions.

• Constant Rule: $y = c \Rightarrow y' = 0$

• Constant Multiple Rule: $h(x) = c f(x) \Rightarrow h'(x) = c f'(x)$

• Sum/Difference Rule: $h(x) = f(x) \pm g(x) \Rightarrow h'(x) = f'(x) \pm g'(x)$

• Product Rule: $h(x) = f(x)g(x) \Rightarrow h'(x) = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $h(x) = \frac{f(x)}{g(x)} \Rightarrow h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $y = f(g(x)) \Rightarrow y' = f'(g(x)) \cdot g'(x)$

Examples

• Product Rule: $y = (4x^2 + 5)e^{x^3 + 3x} \Rightarrow y' = e^{x^3 + 3x}[8x + (4x^2 + 5)(3x^2 + 3)]$

• Quotient Rule: $y = \frac{x^4 - 7}{3 - x^2} \Rightarrow y' = \frac{12x^3 - 2x^5 - 14x}{(3 - x^2)^2}$

• Chain Rule: $y = (5x^9 + 3x^2 - 43)^{34} \Rightarrow y' = 34(45x^8 + 6x)(5x^9 + 3x^2 - 43)^{33}$

Higher Order Derivatives

Higher order derivatives are derivatives of derivatives, useful for analyzing concavity and inflection points.

Order	Notation
First	$f'(x)$, $y'$, $\frac{dy}{dx}$
Second	$f''(x)$, $y''$, $\frac{d^2y}{dx^2}$
Third	$f'''(x)$, $y'''$, $\frac{d^3y}{dx^3}$
n-th	$f^{(n)}(x)$, $y^{(n)}$, $\frac{d^n y}{dx^n}$

• Example: For $f(x) = -2x^2 - 4x - 1$:

  • $f'(x) = -4x - 4$

  • $f''(x) = -4$

  • $f'''(x) = 0$

Critical Points and Classification

Critical points are where the derivative is zero or undefined. They are classified using the second derivative test.

• Steps:

  1. Find $f'(x)$ and $f''(x)$.

  2. Set $f'(x) = 0$ to find critical points.

  3. Use $f''(x)$ to classify:

     • $f''(x^*) < 0$: Maximum

     • $f''(x^*) > 0$: Minimum

     • $f''(x^*) = 0$: Further test needed (possible inflection point)

• Example: $y = x^3 - 3x^2 + 2$ has critical points at $x = 0$ (maximum) and $x = 2$ (minimum).

Applications of Differentiation in Economics

Differentiation is used to analyze marginal changes in economic functions such as revenue, cost, and profit.

• Total Revenue (TR): $TR = p(q) \times q$

• Marginal Revenue (MR): $MR = \frac{d}{dq}(TR)$

• Total Cost (TC): $TC = TFC + TVC$

• Marginal Cost (MC): $MC = \frac{d}{dq}(TC)$

• Profit Maximization: Occurs when $MR = MC$

• Profit ($\pi$): $\pi(q) = TR(q) - TC(q)$

• Break-even Point: Where $TR = TC$

• Elasticity of Demand: $E = -\frac{p}{q} \frac{dq}{dp}$

Examples

• TR and MR: If $p(q) = 100 - q$, then $TR = 100q - q^2$, $MR = 100 - 2q$

• MC: If $TC = 65 + q^{1.5}$, then $MC = 1.5q^{0.5}$; at $q = 25$, $MC = 7.5$

• Profit Maximization: For $TR(q) = 1200q - 2q^2$, $TC(q) = q^3 - 61.25q^2 + 1528.5q + 2000$, maximum profit occurs at $q = 36.5$

• Monopolist Example: For $q = 100 - p$, $TC(q) = \frac{q^3}{3} - 7q^2 + 111q + 50$, maximum profit at $q = 11$, $p = 89$, $\pi(11) = 111.33$

Implicit Differentiation

Implicit differentiation is used when functions are defined implicitly, not explicitly solved for y in terms of x.

• Steps:

  1. Differentiate each term with respect to x.

  2. Treat y as a function of x; apply chain rule when differentiating y terms.

  3. Collect terms involving $\frac{dy}{dx}$.

  4. Solve for $\frac{dy}{dx}$.

• Example: For $x^3 + y^3 + x^2 + 3xy = -1$:

  • $3x^2 + 3y^2 \frac{dy}{dx} + 2x + 3y + 3x \frac{dy}{dx} = 0$

  • Collect $\frac{dy}{dx}$ terms: $\frac{dy}{dx}(3y^2 + 3x) = -(3x^2 + 2x + 3y)$

  • Solve: $\frac{dy}{dx} = \frac{-(3x^2 + 2x + 3y)}{3y^2 + 3x}$