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 nth Derivative of xⁿ Is n Factorial, by Induction

What survives when x^n is differentiated exactly n times, and how does induction turn that guessed pattern into a theorem?

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Your summary note

1. The pattern of repeated differentiation of $x^n$
  Write out $Dx^n = nx^{n-1}$ , $D^2x^n = n(n-1)x^{n-2}$ and $D^3x^n = n(n-1)(n-2)x^{n-3}$ , recording how the exponent and the front factor move at each step.
2. The constant $D^nx^n = n!$
  Carry the count to $D^{n-1}x^n = n(n-1)\cdots 2 \cdot x$ and one step further to $n!$ , then check the small cases $D^2x^2 = 2$ , $D^3x^3 = 6$ , $D^4x^4 = 24$ .
3. Proof by mathematical induction
  Write the base case $Dx = 1 = 1!$ , then the step $D^{n+1}x^{n+1} = D^n((n+1)x^n) = (n+1)D^nx^n = (n+1)n! = (n+1)!$ .
4. One differentiation too many, $D^{n+1}x^n = 0$
  Record the result $D^{n+1}x^n = 0$ together with the one-line argument that settles it.
Attempt 1 of 2