Calculus 1: Derivatives of Functions

• Calculus 1 Module 4 covers derivatives of functions
• The learning objectives are to:
• Define derivatives of a function
• Differentiate functions using the increment method
• Differentiate algebraic functions using the constant, sum, product, and quotient rule
• Find the derivative of an algebraic function using the power formula, chain rule, and general power formula
• Derivatives have early STEM applications

Derivatives in Real Life

• Thrust is the rate of change of momentum

Derivatives of Functions

• Calculus grew out of four major problems European mathematicians worked on in the seventeenth century
• the tangent line problem
• the velocity and acceleration problem
• the minimum and maximum problem
• the area problem
• These problems involve the notion of a limit that calculus can introduce

The Tangent Line Problem

• Partial solutions were from Pierre de Fermat, René Descartes, Christian Huygens, and Isaac Barrows
• The first general solution is credited to Isaac Newton and Gottfried Leibniz
• Newton’s work on the tangent problem stemmed from his interest in optics and light refraction
• For a circle, the tangent line at point P is perpendicular to the radial line at point P and intersects the circle only at the point of tangency
• Finding the tangent line at a point on a curve reduces to finding the slope of the tangent line at that point

Increment

• When a variable x changes from x₁ to x₂, the difference x₂ - x₁ is an increment of x
• In general, an increment of x may be + or - and is denoted by Δx, read as delta x
• Δy denotes the increment of y, and Δf(x) is an increment of f(x)
• The slope of the secant line through points P and Q
• msec = (f(c + Δx) - f(c))/Δx
• As Q moves toward P, that is, as Δx → 0⁺ (or x → c⁺), the secant line PQ approaches the tangent line
• The tangent line is the limiting position of the secant line as Q moves toward P from the right

Definition of the Slope of a Tangent Line to the Graph of f

• If the function f is defined on an open interval containing c, then the slope m of the tangent line to the graph of f at P(c, f(c)) is:
• m = lim (f(c + Δx) - f(c))/Δx = lim (f(x) - f(c))/(x - c) as Δx → 0, as x → c, where Δx = x - c, provided the limit exists

Definition of the Tangent Line to the Graph of a Function

• Supposed the function f is continuous at x = c, the tangent line to the graph of f at the point P(c, f(c)) is defined as:
• The line through P having slope m given by m = lim (f(c + Δx) - f(c))/Δx as Δx → 0, if this limit exists
• The line x = c if lim (f(c + Δx) - f(c))/Δx as Δx → 0⁻ is +∞ or -∞ and lim (f(c + Δx) - f(c))/Δx as Δx → 0⁺ is +∞ or -∞
• The derivative of f at x generalizes the concept of the slope of the tangent line to the graph of f at any x in the domain of f

Definition of the Derivative of a Function

• The derivative of the function f is the function f', such that its value at x in the domain of f is given by:
• f'(x) = lim (f(x + Δx) - f(x))/Δx as Δx → 0, if this limit exists
• If c is a particular value of x ∈ dom f, then:
• f'(c) = lim (f(c + Δx) - f(c))/Δx = lim (f(x) - f(c))/(x - c) as Δx → 0, as x → c, if this limit exists
• Other notations for the derivative of f at x (with respect to x) are:
• dy/dx (Leibniz notation), Dₓ[f(x)], Dₓy, Dₓf(x)
• For a particular value c
• f'(c) is Lagrange notation and dy/dx (x=c) is Leibniz notation

Differentiation

• Differentiation is the process of finding the derivative f' of a function f

Differentiability of a Function

• f is said to be differentiable at x = c if its derivative exists at c
• f is said to be differentiable on an open interval (a, b) if it is differentiable at every point in (a, b)

Two Ways of Finding the Derivative of a Function

• By Increment (4-Step Rule) Method or By Definition
• By Differentiation Formulas (Rules)

Steps in Finding the Derivative by Increment Method

• Given: y = f(x)
• Step 1: Replacing y with y + Δy and x with x + Δx, we get: y + Δy = f(x + Δx)
• Step 2: Solve for Δy: Δy = f(x + Δx) - y = f(x + Δx) - f(x)
• Step 3: Divide Δy by Δx: Δy/Δx = (f(x + Δx) - f(x))/Δx
• Step 4: Find lim (Δy/Δx) as Δx → 0: dy/dx = lim (Δy/Δx) = lim (f(x + Δx) - f(x))/Δx as Δx → 0

Differentiation Formulas (Rules)

• Constant Rule: d/dx (c) = 0, where c is a constant
• Constant Multiple Rule: d/dx (cu) = c(du/dx), where c is constant
• Power Rule: d/dx (xⁿ) = nxⁿ⁻¹, where n ∈ R, n ≠ 0, n ≠ 1
• Sum/Difference Rule: d/dx (u ± v ± w ± ...) = d/dx (u) ± d/dx (v) ± d/dx (w) ± ...
• Product Rule: d/dx (uv) = u(dv/dx) + v(du/dx)
• Quotient Rule: d/dx (u/v) = (v(du/dx) - u(dv/dx))/v²

Other Differentiation formulas

• d/dx (uvw) = uv(dw/dx) + uw(dv/dx) + vw(du/dx)
• d/dx (u/c) = (1/c) (du/dx) where c ≠ 0
• d/dx (c/u) = (-c/u^2) (du/dx) where c ≠ 0
• General Power rule d/dx(u^n) = nu^(n-1) du/dx where n ≠ 0, n ≠ 1
• d/dx (√x) = 1/(2√x)
• d/dx (√u) = 1/(2√u) du/dx
• Chain Rule Formula dy/dx = dy/du * du/dx where y = f(u) & u = g(x)
• Parametric Differentiation where y = f(u), x = g(u) where u is the parameter dy/dx = (dy/du) / (dx/du) = f’(u) / g’(u)

Differentiability and Continuity

• Alternative form of derivative has formula of f’(c) = lim (f(x) - f(c))/(x-c) provided the limit exists
• The existence of a limit in this alternative form requires lim as x → c- and x → c+ of form f(x) − f(c)/x − c exist and are equal
• If f is differentiable at x = c, then f is continuous at x = c
• If a function is not continuous at x = c, it is also not differentiable at x = c