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 Quotient Rule, and the Power Rule for Negative Exponents

Where does the quotient rule's minus sign come from, and how does setting the numerator to 1 stretch the power rule to negative exponents?

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Your summary note

1. The quotient rule $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
  Write the formula down and record the order of the two terms, the minus sign, and the square of the old denominator as the asymmetry to watch.
2. Derivation by the change-and-limit pattern
  Reproduce the derivation: form $\Delta(u/v)$ over the common denominator $(v+\Delta v)v$ , cancel the two copies of $uv$ , divide by $\Delta x$ , and take $\Delta x \to 0$ with $v$ continuous.
3. Reciprocal rule and $\frac{d}{dx}x^{-n} = -nx^{-n-1}$
  Set $u = 1$ to get $\left(\frac{1}{v}\right)' = -\frac{v'}{v^2}$ , then carry $v = x^n$ through the exponent arithmetic to $-nx^{-n-1}$ , and record that the power rule now covers negative $n$ .
Attempt 1 of 2