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∞

Newton and Leibniz Notation, and the Power Rule

How do just two terms of the binomial expansion turn the difference quotient for x^n into the power rule n x^(n-1)?

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Your summary note

1. Notations for the derivative
  Record that $\Delta y = \Delta f$ so the two difference quotients agree, list $f'(x_0)$ , $\frac{df}{dx}$ , $\frac{dy}{dx}$ and $\frac{d}{dx}f$ as names for the same limit, and note that the Leibniz form drops the base point.
2. Binomial expansion needed for $(x+\Delta x)^n$
  Write $(x+\Delta x)^n = x^n + n(\Delta x)x^{n-1} + O((\Delta x)^2)$ and state where the coefficient $n$ and the higher-order junk come from.
3. The power rule $\frac{d}{dx}x^n = nx^{n-1}$
  Form the difference quotient for $x^n$ , cancel $x^n$ against $-x^n$ , divide by $\Delta x$ to get $nx^{n-1} + O(\Delta x)$ , and take the limit as $\Delta x \to 0$ .
4. Extending to polynomials
  State that the rule applies term by term and differentiate one polynomial such as $\frac{d}{dx}(x^3 + 5x^{10}) = 3x^2 + 50x^9$ .
Attempt 1 of 2