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∞

The Derivative of Cosine, and Two Limits at Zero

Why are the two unproved trig limits no accident, but exactly the derivatives of cosine and sine at the single point x = 0?

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Your summary note

1. The cosine sum formula and the same regrouping
  Write $\cos(a+b)=\cos a\cos b-\sin a\sin b$ , expand the difference quotient for $\cos x$ , and pair $\cos x\cos\Delta x$ with $-\cos x$ to recover the brackets $A$ and $B$ .
2. The minus sign, and $\frac{d}{dx}\cos x=-\sin x$
  Record the single structural difference from the sine calculation, the minus sign on the second term, and carry the limit through to the boxed result.
3. $A$ and $B$ as the derivatives at $x=0$
  Work the difference quotients of $\cos x$ and $x=0$ at $x=0$ until they read literally as $A$ and $B$ , and record the values $0$ and $1$ .
4. One rate of change generating all the rest
  State that the derivatives of sine and cosine at every $x$ follow from those at the single point $x=0$ , with the addition formulas supplying every other value.
Attempt 1 of 2