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∞

Higher Derivatives: The Sine Cycle and Three Notations

What makes sine return to itself after four differentiations, and why does the second-derivative symbol put its two 2s in different places?

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Your summary note

1. Higher derivatives as repeated differentiation
  Write the ladder $u'$ , $u''$ , $u'''$ , $u^{(4)}$ in order, and record that each derivative is itself a new function available for differentiating again.
2. The sine cycle back to $\sin x$
  Carry $u=\sin x$ through all four differentiations line by line to $u^{(4)}=\sin x$ , and record that this return is special to sines and cosines, not a general fact.
3. $\frac{d}{dx}$ as an operator, abbreviated $D$
  State what an operator does to a function, then write the chain $u''=\left(\frac{d}{dx}\right)^{2}u=D^2u=\frac{d^2u}{dx^2}$ as one line of equivalent notations.
4. The denominator of $\frac{d^2u}{dx^2}$
  Record the warning that the denominator is the quantity $dx$ squared and is never $d$ of $x^2$ , with nothing at all being done to $x^2$ .
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