/Real Analysis/Differentiation
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Differentiation

Explore derivatives through limits and understand the Mean Value Theorem

ContinuityIntegration

The Derivative as a Limit

The derivative measures instantaneous rate of change. It's defined as a limit of difference quotients:

f'(a) = limh→0 [f(a + h) - f(a)] / h

We'll explore when this limit exists, prove the Mean Value Theorem, and discover the surprising ways that differentiability relates to continuity.

1. The Derivative as a Limit

Watch secant lines approach the tangent line as h → 0. The slope of the secant line converges to the derivative.

Error: |secant slope - f'(a)| = 1.000000

Secant line (current h)
Tangent line (limit as h → 0)

The Derivative as a Limit

The derivative f'(a) is defined as:

f'(a) = limh→0 [f(a + h) - f(a)] / h

Watch as h shrinks: the secant line (connecting two points on the curve) approaches the tangent line (touching at exactly one point).

2. Mean Value Theorem

For a differentiable function on [a, b], there's always a point c where the tangent line is parallel to the secant connecting the endpoints.

a:
b:
MVT Point Found

c = 0.5000

f'(c) = 1.0000

= (f(b) - f(a)) / (b - a) = 1.0000

Endpoints a, b
MVT point c
Secant line
Tangent at c

Mean Value Theorem (MVT)

If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that:

f'(c) = [f(b) - f(a)] / (b - a)

Geometrically: there's a point where the tangent line is parallel to the secant connecting the endpoints.

3. Rolle's Theorem

A special case of MVT: when f(a) = f(b), there must be a point c where f'(c) = 0 (horizontal tangent).

a:
b:

Rolle's Theorem applies!

Rolle's Theorem

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) such that:

f'(c) = 0

This is a special case of MVT: when the secant slope is 0 (horizontal), the tangent must also be horizontal somewhere in between.

4. Differentiability vs Continuity

Differentiability implies continuity, but not vice versa. Explore corners, cusps, and vertical tangents where continuous functions fail to be differentiable.

corner

Left and right derivatives exist but are different

Left derivative: -1, Right: 1

One-sided limits differ!
Left secant
Right secant
Non-differentiable point

Differentiability ⟹ Continuity (but not vice versa!)

If a function is differentiable at a point, it must be continuous there. However, continuity does NOT guarantee differentiability.

• Corner: |x| is continuous but not differentiable at 0

• Cusp: x^(2/3) is continuous but has infinite one-sided derivatives

• Vertical tangent: ∛x is continuous but f'(0) = ∞

• Discontinuity: If not continuous, cannot be differentiable

5. Taylor Polynomials

Higher-order derivatives give us polynomial approximations that match a function's behavior near a point. Watch convergence as degree increases.

Taylor polynomial:

T3(x) = 1.000 +x +0.500x^2 +0.167x^3

Original function f(x)
Taylor polynomial Tn(x)
Expansion center

Taylor Polynomials

The Taylor polynomial of degree n centered at a is:

Tn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + f(n)(a)(x-a)n/n!

As n increases, the polynomial approximates f better near the center. For analytic functions, Tn → f as n → ∞ (within the radius of convergence).

Key Differentiation Theorems

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = [f(b) - f(a)] / (b - a) for some c ∈ (a, b).

Rolle's Theorem

If f(a) = f(b) and f is differentiable on (a, b), then f'(c) = 0 for some c ∈ (a, b).

Differentiability ⟹ Continuity

If f is differentiable at a, then f is continuous at a. The converse is false (e.g., |x| at 0).

Taylor's Theorem

An n-times differentiable function can be approximated by a polynomial plus a remainder term.