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∞

Two Kinds of Formula, and the Derivative of Sine

Why does the sine difference quotient collapse only when its terms are regrouped so a vanishing numerator sits over a vanishing denominator?

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Your summary note

1. Specific formulas versus general formulas
  Record the distinction, listing $\frac{d}{dx}x^n=nx^{n-1}$ and $\frac{d}{dx}(1/x)=-1/x^2$ against $(u+v)'=u'+v'$ and $(cu)'=cu'$ , and note that a polynomial needs both kinds at once.
2. The sine sum formula in the difference quotient
  Write $\frac{\sin(x+\Delta x)-\sin x}{\Delta x}$ and expand the first term by $\sin(a+b)=\sin a\cos b+\cos a\sin b$ with $a=x$ and $b=\Delta x$ , carrying along the trailing $-\sin x$ .
3. The regrouping that keeps a zero over a zero
  Pair $\sin x\cos\Delta x$ with $-\sin x$ , factor out $\sin x$ , write the two brackets $\frac{\cos\Delta x-1}{\Delta x}$ and $\frac{\sin\Delta x}{\Delta x}$ , and note that a $1/0$ term would be fatal.
4. The granted limits $A$ and $B$ , and $\frac{d}{dx}\sin x=\cos x$
  State $(A)\ (\cos\Delta x-1)/\Delta x\to0$ and $(B)\ \sin\Delta x/\Delta x\to1$ , then finish the limit, killing the first term and keeping the factor $\cos x$ .
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