Lesson 1 - Rate of Change and Differentiation Rules

• Variation Function V(h): The absolute change in the dependent variable y = f(x) corresponding to an increment h in the independent variable x: V(h) = f(x1 + h) - f(x1).
• Average Rate of Change A(h): The ratio of variation in function value to the increment h: A(h) = V(h) / h = [f(x1 + h) - f(x1)] / h. Geometrically represents the slope of the secant line passing through points (x1, f(x1)) and (x1+h, f(x1+h)).
• Instantaneous Rate of Change (First Derivative f'(x)): The limiting value of the average rate of change as h approaches zero: f'(x) = lim_{h -> 0} [f(x + h) - f(x)] / h. Represents the slope of the tangent line to the curve at x.
• Differentiability: A function f is differentiable at x = a if the limit definition of the derivative exists and is finite. Differentiability strictly implies continuity at that point.
• Normal Line: The straight line perpendicular to the tangent line at the exact point of tangency. Its slope satisfies: m_normal = -1 / m_tangent (provided m_tangent ≠ 0).

• Limit Definition of the First Derivative (First Principles): [ Formula: f'(x) = dy/dx = lim_{h -> 0} [f(x + h) - f(x)] / h ] Calculates the instantaneous rate of change of f(x) by taking the limit of secant slopes as the distance between points collapses to zero.
• Quotient Rule of Differentiation: [ Formula: d/dx [u / v] = [v · u' - u · v'] / v² ] The derivative of a quotient equals denominator times derivative of numerator minus numerator times derivative of denominator, divided by denominator squared.
• The Chain Rule for Composite Functions: [ Formula: dy/dx = (dy/du) · (du/dx) ; d/dx [u(x)]ⁿ = n [u(x)]^{n - 1} · u'(x) ] Enables differentiation of nested functions by multiplying the derivative of the outer function by the derivative of the inner function.

Differential calculus originated from two classical geometric and physical problems: determining the slope of a tangent line to an arbitrary plane curve (Fermat and Leibniz) and calculating the instantaneous velocity of a moving body (Newton).

Consider a function y = f(x). When x changes from an initial value x1 to x1 + h (where h = Δx is a non-zero increment):

(1) The Variation Function V(h) measures the vertical displacement: V(h) = f(x1 + h) - f(x1).

(2) The Average Rate of Change Function A(h) normalizes this variation per unit increment: A(h) = V(h) / h = [f(x1 + h) - f(x1)] / h. Geometrically, A(h) represents the slope m_secant of the secant line cutting the curve through points P(x1, f(x1)) and Q(x1 + h, f(x1 + h)).

(3) The Instantaneous Rate of Change (The First Derivative dy/dx or f'(x1)) is the limiting behavior as the increment h shrinks towards zero: dy/dx = lim_{h -> 0} A(h) = lim_{h -> 0} [f(x1 + h) - f(x1)] / h. As h -> 0, point Q travels along the curve towards P, and the secant line PQ transitions smoothly into the unique Tangent Line at point P. The derivative equals the tangent slope: f'(x1) = tan θ, where θ is the positive angle made with the positive x-axis.

Evaluating the limit definition for every function is computationally tedious; standard operational differentiation rules are derived directly from first principles:

(1) Constant Rule: The derivative of any constant real number c is zero: d/dx [c] = 0 (since a horizontal line y = c has zero slope).

(2) Power Rule: For any real exponent n: d/dx [xⁿ] = n x^{n - 1}. Examples: d/dx [x] = 1; d/dx [x⁵] = 5x⁴; d/dx [1/x] = d/dx [x⁻¹] = -1 x⁻² = -1/x²; d/dx [√x] = d/dx [x^1/2] = (1/2) x^-1/2 = 1 / (2√x).

(3) Constant Multiple Rule: d/dx [c · f(x)] = c · f'(x).

(4) Sum and Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

(5) Product Rule: The derivative of a product of two differentiable functions: d/dx [u(x) · v(x)] = u(x) · v'(x) + v(x) · u'(x) (First × derivative of second + Second × derivative of first).

(6) Quotient Rule: For rational functions where denominator v(x) ≠ 0: d/dx [u(x) / v(x)] = [v(x) · u'(x) - u(x) · v'(x)] / [v(x)]² (Denominator × derivative of numerator - Numerator × derivative of denominator, divided by denominator squared).

The Chain Rule: If y is a differentiable function of u, and u is a differentiable function of x, then: dy/dx = (dy/du) · (du/dx). In particular, for a function raised to a power: d/dx [f(x)]ⁿ = n [f(x)]^{n - 1} · f'(x).

Tangent and Normal Lines: At any regular point P(x1, y1) lying on the curve y = f(x):

A. Tangent Line Equation: Calculate slope m = f'(x1). The tangent line equation is: y - y1 = m(x - x1). If m = 0, the tangent is horizontal (y = y1). If the denominator of f'(x1) is zero, the tangent is vertical (x = x1).

B. Normal Line Equation: Because the normal line is perpendicular to the tangent, its slope is m_normal = -1 / m. The normal line equation is: y - y1 = (-1 / m)(x - x1).

Kinematic Physical Applications: If s(t) represents the position vector of a particle at time t: (1) Instantaneous Velocity v(t) = ds/dt = s'(t). (2) Instantaneous Acceleration a(t) = dv/dt = d²s/dt² = s''(t). When a particle changes its direction of motion, it comes to instantaneous rest: v(t) = 0.

• Exam Secret: 'Continuity does NOT imply differentiability'. A classic example is f(x) = |x| at x = 0: it is continuous, but the left derivative is -1 while the right derivative is +1. Because left and right limits do not match, f'(0) does not exist (sharp corner).
• Exam Trap: If the tangent line makes an obtuse angle with the positive x-axis, its slope m is negative (m = tan θ < 0). If θ = 90°, m is undefined (vertical tangent).
• When finding the rate of change of area or volume with respect to time, always apply the Chain Rule: dA/dt = (dA/dr) · (dr/dt).

• Economic Marginal Analysis: In macroeconomics, marginal cost MC(x) = C'(x) is the derivative of the total cost function, indicating the cost of manufacturing one additional unit.
• Aerospace Trajectory Optimization: Rocket guidance computers calculate instantaneous rates of velocity change to adjust propellant throttle and maintain stable orbital insertion.
• Civil Engineering Stress Analysis: Structural deflection of suspension bridges under vehicle loads is modeled using fourth-order differential equations derived from curvature slopes.