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∞

Infinite Discontinuities: 1/x and Its Derivative

Why is writing lim(x to 0) 1/x = infinity simply wrong, and why does the graph of a derivative look nothing like the graph it came from?

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Your summary note

1. One-sided limits of $1/x$ at $0$
  Write $\lim_{x\to 0^+}\frac{1}{x}=+\infty$ and $\lim_{x\to 0^-}\frac{1}{x}=-\infty$ , and record that $\lim_{x\to 0}\frac{1}{x}=\infty$ is wrong since the two sides go opposite directions.
2. Graph of the derivative $f'(x)=-\frac{1}{x^2}$
  Draw $-1/x^2$ under $1/x$ , track the slope on both branches read left to right, and show it is negative everywhere with $\lim_{x\to 0}-\frac{1}{x^2}=-\infty$ from both sides.
3. Derivative of an odd function is even
  State that differentiating the odd function $1/x$ gives the even function $-1/x^2$ , and connect this to $x^{-1}$ being an odd power and $x^{-2}$ an even power.
4. Ugly discontinuities: $\sin\frac{1}{x}$
  Record that $\sin\frac{1}{x}$ oscillates infinitely often as $x\to 0$ with no one-sided limit, so the limit does not exist and it is neither jump, removable, nor infinite.
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