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∞

Finishing the Proof, and the Product and Quotient Rules

What does rotating an angle by ninety degrees reveal about the little triangle, and why does that finish the geometric proof?

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Your summary note

1. The unknown angle $\angle QPR$
  State the remaining step of the argument: the hypotenuse together with the angle at $P$ determines the vertical side $PR$ of the small right triangle.
2. The ninety-degree rotation argument
  Record the two geometric facts, $PQ$ approaching the tangent perpendicular to $OP$ and $PR$ vertical, then carry $\theta$ onto the angle at $P$ by rotation to get $\angle QPR \approx \theta$ .
3. The conclusion $\Delta y \approx \Delta\theta \cos\theta$
  Finish the computation: write $PR \approx \Delta\theta\cos\theta$ , divide by $\Delta\theta$ , pass to the limit, and state the geometric result $\frac{d}{dx}\sin x = \cos x$ .
4. The product rule and the quotient rule
  Write $(uv)' = u'v + uv'$ and $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ for $v \neq 0$ , and note the change-one-factor-at-a-time way of remembering the first.
Attempt 1 of 2