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∞

Tangents to 1/x and the Triangle of Area 2

Why does every tangent to y = 1/x cut off a triangle of area exactly 2, no matter where along the curve you draw it?

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Your summary note

1. Setup and the single calculus step
  Draw the first-quadrant hyperbola with a tangent cutting the axes, label the point $(x_0,y_0)$ , and write the tangent line $y-y_0=\frac{-1}{x_0^2}(x-x_0)$ as the whole calculus content.
2. The intercepts $x=2x_0$ and $y=2y_0$
  Carry out the algebra of setting $y=2y_0$ with $y_0=1/x_0$ to get $x=2x_0$ , then get $y=2y_0$ from the exchange symmetry $y=1/x \iff xy=1 \iff x=1/y$ , noting $x=0$ works too.
3. Constant area $=2$
  Compute $\frac{1}{2}(2x_0)(2y_0)=2x_0y_0=2$ , record that the answer is independent of $x_0$ , and note that $y=c/x$ gives constant area $2c$ .
4. Variable bookkeeping as the real difficulty
  Record that $x$ , $y$ , $x_0$ , $y_0$ all coexist here and that $y$ deliberately means the curve in $y_0=1/x_0$ but the horizontal line in $y=0$ .
Attempt 1 of 2