Week 5 Notes: Derivatives, Limits, Inverses, and Applications
About Functions and Local Linearity
Graphs of functions are not generally straight lines; “well-behaved” functions are locally close to straight lines when you zoom in.
Example intuition: for f(x) = x^2, the tangent line at a point shows how the function behaves locally around that point.
Tangent context for f(x) = x^2 at a = 0.5:
Point on curve: (0.5, 0.25)
Tangent line: y = x - 0.25 (this is the tangent line) and the slope is f'(0.5) = 1.
Other nearby lines: secant lines with slopes 1.1, 1.2, 1.3, 1.4 given in the material: y = (m)x + b, e.g., for learning about how secants approximate the tangent.
Derivative serves as the instantaneous rate of change and the slope of the tangent line to the graph of a function.
Derivative Notation and Meaning
For a function f:\,R o R, define the derivative as the function f':R o R with
f'(x) = \dfrac{dy}{dx} = \dfrac{df}{dx} where y = f(x).
Derivative interpretation:
The instantaneous rate of change of f at a point.
Measures how fast the function is changing for a marginal change in the input, typically in x.
Indicates local stability: increasing, decreasing, or stationary (when derivative is zero).
Derivative of Power (and General) Functions
For a power function f(x) = x^a with suitable domain, the derivative is
f'(x) = a x^{a-1}
For integer powers, this specializes to
\dfrac{d}{dx} x^n = n x^{n-1} (valid for any real n with appropriate domain)
Example behavior: plotted relationships show how slope changes with x; derivative grows with x for positive a.
Special case: For a function defined on [0, +\infty), the derivative at any x>0 is a x^{a-1}; examples illustrate the derivative curve alongside the original function.
Derivative Rules: Linearity, Product, and Quotient
Linearity (Rule 1):
If g and w are differentiable, then for all real a:
(g + w)'(x) = g'(x) + w'(x)
(ag)'(x) = a g'(x)
Product Rule:
If f,g are differentiable, then
(fg)'(x) = f(x) g'(x) + f'(x) g(x)
Quotient Rule:
If f,g differentiable and g(x) \neq 0, then
\left(\dfrac{f}{g}\right)'(x) = \dfrac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}
Derivatives at a point (local rules):
For a point a\in\text{dom}(f), the derivative of the product and quotient evaluated at a are
(fg)'(a) = f(a) g'(a) + f'(a) g(a)
(f/g)'(a) = \dfrac{f'(a) g(a) - f(a) g'(a)}{[g(a)]^2}
These rules enable derivative computations of many composite expressions.
Existence of the Derivative and Examples of Non-Differentiability
Not all functions are differentiable everywhere; some are differentiable everywhere (e.g., polynomials, sine, exponential on their domains).
Absolute value function has a kink at 0 and is not differentiable there.
Graphically: kink at (0,0) implies no tangent line in the usual sense at x = 0.
Function h(x) = x^{1/3} is differentiable for all x \neq 0 with
h'(x) = \dfrac{1}{3 x^{2/3}}
But at x=0 the derivative does not exist (the tangent is vertical at 0).
A warning: the derivative describes local behavior; globally a function can behave very differently outside a neighborhood where the derivative exists.
Example with g(x) = 1/x (domain \mathbb{R} \setminus {0}):
g'(x) = -1/x^2 = -x^{-2}
Here both the function and derivative are defined on the same domain, but the derivative is negative everywhere in the domain, so the function is locally decreasing on its domain. (Note: the domain excludes 0.)
Cost Functions and Economic Applications
Cost decomposition:
Total cost: C(q) = FC + VC(q) where
FC = C(0) is fixed cost (constant, does not depend on q), and
VC(q) = C(q) - C(0) is the variable cost dependent on output level q.
Marginal cost function:
The marginal cost is the derivative of total cost: C'(q).
Interpretation: how much more it costs to increase production by one unit when currently producing q units.
Consequences of cost decomposition:
Since FC' = 0, C'(q) = VC'(q); economists focus on marginal cost rather than marginal fixed cost or marginal variable cost separately.
Other related cost measures:
Average cost: AC(q) = \dfrac{C(q)}{q} (for q>0)
Average fixed cost: AFC(q) = \dfrac{FC}{q}
Average variable cost: AVC(q) = \dfrac{VC(q)}{q}
Relationship: C(q) = q \, AC(q) and AC(q) = AFC(q) + AVC(q)
Derivatives of average cost and critical points:
If AC'(q) > 0, then since AC(q) = \dfrac{C(q)}{q},
(AC'(q) > 0) \iff \dfrac{q C'(q) - C(q)}{q^2} > 0 \iff C'(q) > \dfrac{C(q)}{q} = AC(q)
If AC'(q) = 0, then
(q C'(q) - C(q))/q^2 = 0 \iff q C'(q) = C(q) \iff C'(q) = AC(q)
If AC'(q) < 0, the inequality reverses accordingly (noting the same algebraic form).
Important interpretation: the derivative of total cost relative to output interacts with average cost to determine cost behavior and optimal scales.
Geometry, Sign, and Critical Points of Derivatives
The derivative at a point is the slope of the tangent line to the graph at that point; a geometric interpretation.
Sign of the derivative:
If f'(a) > 0, then f is locally increasing around a.
If f'(a) < 0, then f is locally decreasing around a.
If f'(a) = 0, then x=a is a stationary/critical point. A critical point may be a local max, local min, or neither.
Parabola example: for g(x) = x^2 + c, g'(x) = 2x, so g'(0) = 0 (critical point at the vertex).
Examples of critical-point behavior:
f(x) = -x^2 has a local maximum at x=0 with derivative f'(x) = -2x and f'(0) = 0.
g(x) = x^2 has a local minimum at x=0 with derivative g'(x) = 2x and g'(0) = 0.
h(x) = x^3 has derivative h'(x) = 3x^2, so h'(0)=0 but it has neither a local max nor min at 0.
Special note: for cubic root function h(x) = x^{1/3}, derivative at 0 is not finite; the derivative is undefined at 0 despite the point being a critical point for other powers.
Inverse Functions and Inverses
Inverse definition: For two functions g: B \to A and f: A \to B, they are inverses if
(g \circ f)(a) = a for all a \in A and (f \circ g)(b) = b for all b \in B.
A function f: A \to B has an inverse iff it is both injective and surjective (bijective).
Notation: the inverse is denoted f^{-1}: B \to A, which swaps the domain and codomain.
Simple inverse for a linear function: if f(x) = m x + b with m \neq 0, then
f^{-1}(y) = \dfrac{y - b}{m}.
Example: f(x) = 2x - 3\Rightarrow f^{-1}(x) = \dfrac{x+3}{2}.
Properties of inverse derivatives (Inverse Function Theorem):
If f: \mathbb{R} \to \mathbb{R} has an inverse and both f and f^{-1} are differentiable, and if b = f(a), then
f'(a)\,(f^{-1})'(b) = 1.
Consequently, if f'(a) > 0 then $(f^{-1})'(b) > 0$, and if f'(a) < 0 then $(f^{-1})'(b) < 0$.
Inverse Function Theorem in practice: for common functions like f(x) = e^x and g(y) = \ln y, the relationship $(e^x)' (\ln y)'(y_0) = 1$ can be used to derive the derivative of the inverse relationship.
Inverse of composite functions interacts with chain rule (see below).
A note on special identities: for the exponential sequence approaching Euler’s number e\approx 2.71828…, the sequence $a_n = 1 + 1/n)^n$ converges to e as shown in the related slides.
Composite Functions and the Chain Rule
Composite functions: for f: \mathbb{R} \to \mathbb{R} and g: \mathbb{R} \to \mathbb{R},
(f \circ g)(x) = f(g(x))
(g \circ f)(x) = g(f(x))
The order of composition matters.
Inverses of composites: If both f and g have inverses, then
(f \circ g)^{-1} = g^{-1} \circ f^{-1}.
Chain Rule (basic form): If f: \mathbb{R} \to \mathbb{R} and g: \mathbb{R} \to \mathbb{R} are differentiable, then
(f \circ g)'(x) = f'(g(x))\, g'(x).
Examples:
If p(x) = e^{3x}, then p'(x) = 3 e^{3x}.
If h(x) = (x^5 + e^{3x})^2, then
{h'(x) = 2 (x^5 + e^{3x}) (5x^4 + 3e^{3x})}.
If q(x) = \ln(7x), then
q'(x) = \dfrac{1}{x}.
The same chain rule applies to more complex compositions.
Exponential and logarithmic derivatives:
For exponential base b > 0, F(x) = b^x = e^{(\ln b) x}, so F'(x) = (\ln b)\, b^x.
For natural log, \dfrac{d}{dx} \ln x = \dfrac{1}{x} for x > 0.
Exponential function special case: for f(x) = e^x, f'(x) = e^x, i.e., the function equals its own derivative.
The Derivative of Ratios and Logs
Ratios: If h(x) = \dfrac{f(x)}{g(x)}, then
h'(x) = \dfrac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}.
Equivalently, h'(x) = \dfrac{f'(x)}{g(x)} - \dfrac{f(x) g'(x)}{[g(x)]^2}.
Also, \dfrac{h'(x)}{h(x)} = \dfrac{f'(x)}{f(x)} - \dfrac{g'(x)}{g(x)}.
Example pattern: If you take specific f, g and form h = f/g, you can compare the growth rates via the logarithmic derivative identity above.
Second and Higher Order Derivatives
Second derivative: If y = f(x), then the first derivative is f'(x) = \dfrac{dy}{dx} and the second derivative is
f''(x) = \dfrac{d^2 y}{dx^2} = \dfrac{d}{dx} (f'(x)).
Notation for higher derivatives: the n-th derivative is denoted
f^{(n)}(x) = \dfrac{d^n y}{dx^n} = \dfrac{d^n f}{dx^n}.
Examples show several polynomial and exponential functions with their first, second, and higher derivatives.
Geometric interpretation:
If f''(x) > 0, then the first derivative f'(x) is strictly increasing near x, i.e., the function is strictly convex there.
If f''(x) < 0, then the function is strictly concave there.
Affine functions: If f(x) = m x + b, then f''(x) = 0, so they are both weakly convex and weakly concave (neither strictly).
Implicit Differentiation
When an equation relates x and y implicitly, you can differentiate implicitly to obtain dy/dx.
Example: the circle equation x^2 + y^2 = 1 differentiates to
2x + 2y\dfrac{dy}{dx} = 0 \Rightarrow \dfrac{dy}{dx} = -\dfrac{x}{y}.
Example: If xy^3 + 2x^2 y = 3x - y, implicit differentiation yields a solution for \dfrac{dy}{dx} in terms of x and y by isolating dy/dx on one side.
A common technique: differentiate both sides with respect to x, then solve for dy/dx.
Trigonometry: Tangent Orthogonality and Complements
Tangent axis in the unit circle:
The tangent is not defined at angles \alpha = k\pi + \frac{\pi}{2} for integers k.
Sign of tangent depends on the quadrant: tan(α) > 0 in quadrants I and III; tan(α) < 0 in II and IV.
Complement angle identity:
\tan \alpha \tan (90^{\circ} - \alpha) = 1.
Also \tan(90^{\circ} - \alpha) = \dfrac{1}{\tan \alpha} (when defined).
Inverse Function Theory and Examples
Inverse when: a function and its inverse swap domain and codomain and satisfy
(f^{-1} \circ f)(a) = a, \quad (f \circ f^{-1})(b) = b.
Existence of inverse requires bijectivity: injective and surjective.
Finding inverses:
For linear affine: f(x) = m x + b, \quad m \neq 0, the inverse is f^{-1}(y) = \dfrac{y - b}{m}.
Inverse derivative relationship (IFT): If f has an inverse and both are differentiable, then at the corresponding points b = f(a),
f'(a) \,(f^{-1})'(b) = 1.
Global and Local Properties: Maxima, Minima, and Existence
Global maximum: A point x is a global maximizer for a function f if f(x) > f(z) for every z in the domain.
Example: f(x) = -x^2 has global maximum at x = 0 with f(0) = 0.
Global minimum: A point x is a global minimizer if f(x) < f(z) for all z in the domain.
Example: f(x) = x^2 has global minimum at x = 0 with f(0) = 0.
Existence and uniqueness: A function may lack a global max or min, or may have multiple maximizers/minimizers.
The existence of a global max/min depends on the domain and the form of the function.
Real-World Applications: Macroeconomic Model (Output and Capital)
Model setup:
Product (output): Y = f(K) L with $f(0)=0$ and increasing marginal returns implicitly assumed.
Capital dynamics: K = \delta K + I with a constant depreciation rate 0<\delta<1.
Investment: I = Y - C.
Consumption: C = (1 - s) Y with savings rate 0 < s < 1.
From these: 1 - \delta = \text{depreciation rate} and s = \text{savings rate}.
Equation for the capital-labor balance:
Using the identity K = \delta K + I, rearrange to (1 - \delta)K = I and substitute for I and C to obtain the relation
(1 - \delta) K = s Y = s f(K) L.
Solving for K often proceeds by defining
g(K) = \dfrac{K}{f(K)} and the equation becomes
g(K) = \dfrac{s L}{1 - \delta}.
If $g'(K) > 0$ for all K, then $g$ is increasing, so by the Inverse Function Theorem, its inverse $g^{-1}$ is increasing, giving
K = g^{-1}\left( \dfrac{s L}{1 - \delta} \right).
Implications:
Since both $g$ and $g^{-1}$ are increasing, larger values of $s$, $L$, or smaller depreciation (larger $1 - \delta$) increase $K$ and hence increase $Y = f(K)L$.
Limits of Functions and Continuity
Limit concept: The limit of f(x) as x\to a is the value L such that f(x) gets arbitrarily close to L when x is arbitrarily close to a (but not necessarily equal to a).
Notation: \lim_{x \to a} f(x) = L.
The value f(a) is irrelevant to the limit.
Examples:
\lim_{x\to 3} (2x+1) = 7.
For a function with a removable discontinuity, e.g., f(x) = \dfrac{x^2 - 4}{x - 2} for x \neq 2, we have
\lim_{x\to 2} f(x) = 4 even though the function is not defined at x=2 (since the expression simplifies to x+2 for x\neq 2).
Continuity:
A function f: \mathbb{R} \to \mathbb{R} is continuous at a if
f(a) = \lim_{x\to a} f(x).
Continuity everywhere (global): A function is continuous on its domain if it is continuous at every point of its domain.
Examples of continuous families: polynomials, exponentials, affine functions, and many others.
The Dirichlet-type function that is 1 on rationals and 0 on irrationals is not continuous at any point.
Theorems on continuity:
If f and g are continuous at a, then
f + g, f - g, and f g are continuous at a.
If g(a) \neq 0, then \dfrac{f}{g} is continuous at a.
Injective, Surjective, and Surjective-Injective (Bijective) Functions and Pigeonhole Principle
Injective (one-to-one): A function f: D \to C is injective if, for all a,b \in D, a \neq b\Rightarrow f(a) \neq f(b). No two domain points map to the same codomain point.
Examples:
The affine function f(x) = mx + b is injective iff m \neq 0.
The function f(x) = x^2 is not injective on the whole real line, since (-7)^2 = 7^2; but on [0, \infty) it is injective.
Not a function: When a rule assigns more than one value to a single input (e.g., a vertical line in a graph that hits the domain point with multiple outputs), which is not allowed by the definition of a function.
Surjective (onto): A function f: A \to B is surjective if for every b \in B there exists an a \in A such that f(a) = b; equivalently, the range of $f$ equals the codomain: f(A) = B.
Range relationships and examples:
Some functions are injective but not surjective, some are surjective but not injective, some are both (bijective), and some are neither.
Finite domain/codomain (Pigeonhole Principle):
If A and B are finite and |A| > |B|, there is no injective function f: A → B.
If there is an injective function f: A → B, then |A| ≤ |B|.
There exists a surjective function f: A → B iff |A| ≥ |B|.
There is a function f: A → B that is both injective and surjective (bijective) iff |A| = |B|, in which case an inverse function exists.
Important consequence: If |A| = |B|, not every function is bijective; bijectivity depends on the rule, and having a bijection is equivalent to the existence of an inverse.
Fundamentals of the Tangent, Inverses, and Limits in Trigonometry
Tangent axis and positivity:
Tangent is defined for most angles except at odd multiples of 90 degrees where it blows up to infinity.
The sign of the tangent depends on the angle's quadrant.
Complement identities:
Tangent of a complement: \tan(90^{\circ}-\alpha) = \dfrac{1}{\tan \alpha} when defined.
The tangent function is not defined at angles \alpha = k\pi + \frac{\pi}{2} for integers k.
Limits, Continuity, and Derivative Relationships (Summary of Key Theorems)
Limit: \lim_{x \to a} f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a (x ≠ a).
Continuity: A function is continuous at a if f(a) = \lim_{x \to a} f(x).
Continuity rules: If f and g are continuous at a, then
f+g,\, f-g,\, fg are continuous at a; and if g(a) \neq 0, then f/g is continuous at a.
Examples of continuous functions: affine, polynomial, exponential; while some non-continuous constructions exist (e.g., Dirichlet-type function).
Remarks on the First and Higher Derivatives: Global and Local Behaviors
The derivative provides local behavior; global behavior can differ (e.g., a function with a linear tangent can still have global nonlinearity).
Second-order and higher derivatives measure curvature and concavity/convexity:
If f''(x) > 0, the function is locally convex (curving upward).
If f''(x) < 0, the function is locally concave (curving downward).
Summary of Special Functions and Derivative Rules
Exponential functions:
For base b>0, F(x) = b^x = e^{(\ln b) x} and F'(x) = (\ln b) b^x.
Special case: F(x) = e^x has derivative F'(x) = e^x.
Logarithmic function:
f(x) = \ln x, \quad x>0, and f'(x) = \dfrac{1}{x}.
Derivatives of a ratio and its logs:
As above, the ratio rule and the logarithmic derivative relations provide compact tools for handling quotients.
Additional Topics for Mastery (Selected Commands and Identities)
Tricky exponent identities:
For a base b > 0, b^n = e^{(\ln b) n}, so derivatives/equalities can be analyzed via exponentials.
Chain Rule (extended view): for f( g(x) ), you repeatedly apply the chain rule to composed layers.
Global maxima/minima existence depends on domain and function behavior at infinity; not all functions have global extrema.
Important: the derivative gives local slope information, not necessarily global trend or absolute extrema.
For composite inverses: if both f and g have inverses, then the inverse of the composite is the composition of the inverses in reverse order: (f\circ g)^{-1} = g^{-1} \circ f^{-1}.