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∞

Why the Trig Derivatives Hold Only in Radians

When two quantities vanish together, what decides whether their ratio tends to 1 or to 0 — and why does measuring in degrees wreck it?

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Your summary note

1. A limit as a comparison of rates
  Record that both the numerator and the denominator vanish, and state the criterion that settles such a limit: which quantity vanishes faster than the other.
2. The two ratios $A$ and $B$ contrasted
  Write the chord and the arc as two vanishing quantities that stay comparable, giving ratio $1$ , against the bowstring gap dying far faster than the arc, giving $0$ .
3. The zoom-and-rescale picture
  Describe the process as shrinking $\theta$ while blowing the figure back up, and record that the rescaling preserves only the ratio, not the absolute sizes.
4. Radians as the required unit of angle
  State that the proof of $B$ measured $\theta$ as arc length on the unit circle, and record that $\frac{d}{dx}\sin x = \cos x$ and $\frac{d}{dx}\cos x = -\sin x$ fail in degrees.
Attempt 1 of 2