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∞

A Geometric Proof: Sine as a Height on the Circle

How does watching a particle climb the unit circle turn the derivative of sine into a question about one tiny right triangle?

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Your summary note

1. The particle at $P$ on the unit circle
  Set up the second proof, valid at every $\theta$ rather than only at $\theta = 0$ , with a particle at $P$ whose height is $y = \sin\theta$ .
2. The triangle $PQR$ and $\Delta y = PR$
  Draw $Q$ at angle $\theta + \Delta\theta$ , drop the perpendicular from $Q$ to the vertical line through $P$ at $R$ , and record $\Delta y = PR$ , not the chord $PQ$ .
3. The difference quotient $\Delta y / \Delta\theta$
  Write the ratio $\frac{\Delta y}{\Delta\theta}$ for $y = \sin\theta$ and identify its limit as the quantity being computed, the derivative of sine.
4. The hypotenuse of length essentially $\Delta\theta$
  Record the principle that short pieces of curve are nearly straight, the approximation arc $PQ \approx$ segment $PQ$ , and the exact arc length $\Delta\theta$ on the unit circle.
Attempt 1 of 2