3.1.1 Intro to Derivatives

Fundamental concept of calculus; measures an instantaneous rate of change at a single point.
Allows us to analyze systems in perpetual change – physics, economics, biology, engineering, etc.
Geometric picture: slope of a tangent line to a curve at a point.
- Positive derivative → rising tangent; negative derivative → falling tangent.
Connects directly to Chapter 2 material on limits; derivative exists only if the relevant limit exists.

Average velocity / slope (secant line) on interval [a,b]:
- \text{Average Rate} = \frac{f(b)-f(a)}{b-a}
Instantaneous velocity / slope (tangent line) at x=a:
- \displaystyle f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a} ("limit-of-the-quotient" definition)
- Equivalent increment ((h)-form) definition:
  \displaystyle f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}
Secant line simply joins two points; as the two points merge, the secant approaches the tangent (visual limit concept).

Picture two nearby points sliding together along the curve.
Secant slope gradually morphs into tangent slope.
Tangent line is unique; cannot draw two distinct tangent lines through the same point of a smooth curve.

Setting: Rock launched vertically with initial velocity 96\,\text{ft/s}.
Height function: s(t) = -16t^{2}+96t (feet) – derived from kinematics with gravitational constant -32\,\text{ft/s}^{2}.
Given point P(1,80) (after 1 s, rock is 80 ft high).

Goal A: Instantaneous velocity at t=1

Use definition \displaystyle v(1)=\lim_{t\to1}\frac{s(t)-s(1)}{t-1}.
Compute s(1)=-16(1)^{2}+96(1)=80.
Algebraic simplification:
- Factor numerator, cancel t-1.
- Cancel & evaluate limit → 64.
Result: v(1)=64\,\text{ft/s} (also the tangent slope).

Goal B: Equation of the tangent line

Point–slope form with point P(1,80) and slope 64:
y-80 = 64\,(t-1) → y = 64t + 16.
Interpretation: linear approximation of height near t=1.

Limit formula: \displaystyle f'(2)=\lim_{x\to2}\frac{\dfrac{3}{x}-\dfrac{3}{2}}{x-2}.
Steps
1. Combine fractions to obtain single numerator \frac{6-3x}{2x}.
2. Factor -3( x-2); cancel x-2 with denominator.
3. Remaining expression \displaystyle -\frac{3}{2x}.
4. Evaluate at x=2 ⇒ -\tfrac34.
Tangent line: slope m=-\tfrac34, point \bigl(2,\tfrac32\bigr).
- y-\tfrac32=-\tfrac34\,(x-2) ⇒ y=-\tfrac34x+3.

Function: f(x)=x^{3}+4x. Find tangent at x=1.

Apply h-definition: \displaystyle f'(1)=\lim_{h\to0}\frac{f(1+h)-f(1)}{h}.
Compute expansion:
- f(1+h)=(1+h)^3+4(1+h)=1+3h+3h^{2}+h^{3}+4+4h
- Simplify → h^{3}+3h^{2}+7h+5.
Subtract f(1)=5 → numerator h^{3}+3h^{2}+7h.
Factor h, cancel with denominator h.
Evaluate limit h→0 ⇒ 7.
Tangent slope 7; point \bigl(1,5\bigr).
- Equation: y-5=7(x-1) \Rightarrow y=7x-2.

Whenever direct substitution in the limit produces 0/0, algebra (factoring, rationalizing, common denominators) is required to expose and cancel the problematic factor.
Differentiability hinges on limit existence; if left & right limits mismatch or blow up, tangent line doesn’t exist.
Physical relevance: instantaneous velocity, marginal cost, growth rates, electric current at a moment, etc.
Philosophically, derivative formalizes the intuitive idea of "change at an instant"—a concept that puzzled thinkers from Zeno to Newton.