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 Chain Rule: Differentiating a Function of a Function

Why is the derivative of a composition a product, and what has to happen to the inside variable for the cancellation to survive the limit?

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Your summary note

1. Composition as substitution, with a named intermediate variable
  Write $(\sin t)^{10}$ as two steps, $x = \sin t$ then $y = x^{10}$ , note the $\sin^{10} t$ shorthand, and record that the last step substitutes $x$ back.
2. The one-line derivation of $dy/dt = (dy/dx)(dx/dt)$
  Write $\Delta y/\Delta t = (\Delta y/\Delta x)(\Delta x/\Delta t)$ , cancel $\Delta x$ , record the limit step as $\Delta t\to0$ , and state both $dy/dt = (dy/dx)(dx/dt)$ and $\frac{d}{dt}f(g(t)) = f'(g(t))g'(t)$ .
3. Worked example $(\sin t)^{10}$
  Take $dy/dx = 10x^9$ and $dx/dt = \cos t$ , multiply them, and substitute $x = \sin t$ back to finish at $10\sin^9 t\cos t$ .
4. $\sin(10t)$ and the shortcut without the middle variable
  With $x = 10t$ and $y = \sin x$ , reach $10\cos(10t)$ , then redo it without naming $x$ , differentiating the outside function, leaving the inside expression alone, and multiplying by its derivative.
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