Single Variable Calculus

Differentiation

01The Derivative as the Slope of the Tangent Line 02Secant Lines and the Limit Definition of the Derivative 03Differentiating 1/x Straight From the Definition 04Tangents to 1/x and the Triangle of Area 2 05Newton and Leibniz Notation, and the Power Rule Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01The Derivative as an Instantaneous Rate of Change 02Rates of Change Without Time: Gradient and Sensitivity 03Easy Limits, 0/0, and One-Sided Limits 04Continuity, Jumps, and Removable Discontinuities 05Infinite Discontinuities: 1/x and Its Derivative 06Differentiable Implies Continuous Problem set0/10 Problem set 20/10 Practice∞

01Two Kinds of Formula, and the Derivative of Sine 02The Derivative of Cosine, and Two Limits at Zero 03Bow and Bowstring: Proving the Two Trig Limits 04Why the Trig Derivatives Hold Only in Radians 05A Geometric Proof: Sine as a Height on the Circle 06Finishing the Proof, and the Product and Quotient Rules Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01Proving the Product Rule: Change One Factor at a Time 02The Quotient Rule, and the Power Rule for Negative Exponents 03The Chain Rule: Differentiating a Function of a Function 04Higher Derivatives: The Sine Cycle and Three Notations 05The nth Derivative of xⁿ Is n Factorial, by Induction Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01The Power Rule for Rational Exponents 02The Slope of a Circle, Two Ways 03Implicit Differentiation of a Quartic Curve 04Inverse Functions and the Reflection Across y = x 05Derivatives of the Arctangent and Arcsine Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Differentiable Implies Continuous

How can multiplying by (x - x0)/(x - x0) prove that differentiability forces continuity without ever dividing by zero?

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Your summary note

1. Restating continuity at $x_0$
  State the theorem, then write continuity $\lim_{x \to x_0} f(x) = f(x_0)$ in the equivalent form $\lim_{x \to x_0}(f(x) - f(x_0)) = 0$ that the proof checks.
2. The one-line proof
  Reproduce $\lim_{x \to x_0}(f(x) - f(x_0)) = \lim_{x \to x_0}\left[\frac{f(x) - f(x_0)}{x - x_0}\right](x - x_0) = f'(x_0) \cdot 0 = 0$ , noting the first factor's limit is where differentiability is used.
3. Not division by zero
  Record that multiplying and dividing by $x - x_0$ is legal since the limit excludes $x = x_0$ , so $x - x_0$ is small but never zero, and note this supports the product and quotient rules.
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