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 Power Rule for Rational Exponents

How does turning y = x^(m/n) into y^n = x^m let the chain rule prove the power rule for every rational exponent?

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Your summary note

1. Setting up $y^n = x^m$ and differentiating it
  Write $y = x^{m/n}$ , raise it to the $n$ th power to get $y^n = x^m$ , note which equation gets differentiated, and apply the chain rule to reach $ny^{n-1}\frac{dy}{dx} = mx^{m-1}$ .
2. Solving for $\frac{dy}{dx}$ and the exponent arithmetic
  Divide to isolate $\frac{dy}{dx}$ , substitute $y = x^{m/n}$ back in, and carry out the exponent algebra $(m-1) - \frac{m(n-1)}{n} = \frac{m}{n} - 1$ to land on $ax^{a-1}$ .
3. The case $m = 1$ and the $n$ th roots
  Record that $a = \frac{m}{n}$ with $m = 1$ gives $x^{1/n}$ , and differentiate one root such as $\sqrt{x}$ or $x^{1/3}$ as the worked example.
Attempt 1 of 2