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 as the Slope of the Tangent Line

Why can a hand draw the tangent line instantly, and what exactly must calculus supply so a machine could do it too?

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Your summary note

1. The geometric problem of the tangent line
  State the task: find the tangent line to $y = f(x)$ at a point $(x_0, y_0)$ , and note the goal of turning the hand-drawn line into an analytic procedure a machine could carry out.
2. Point-slope equation and the two pieces of information
  Write $y - y_0 = m(x - x_0)$ and record that a tangent line needs the height $y_0 = f(x_0)$ , which is not calculus, and the slope $m$ , which is.
3. Definition of $f'(x_0)$
  State that $f'(x_0)$ , the derivative of $f$ at $x_0$ , is the slope of the tangent line to $y = f(x)$ at $P$ , and note this names but does not yet compute the number.
Attempt 1 of 2