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∞

Proving the Product Rule: Change One Factor at a Time

Why does differentiating a product hand back a sum, and what does adding and subtracting one middle term reveal about the change in uv?

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Your summary note

1. The product rule $(uv)' = u'v + uv'$
  Write the formula and state in words which factor is differentiated in each term, recording that differentiating a product returns a sum of two products.
2. Worked example $\frac{d}{dx}(x^n \sin x)$
  Name $u = x^n$ and $v = \sin x$ , list $u'$ and $v'$ , and carry the rule through to $nx^{n-1}\sin x + x^n\cos x$ ; longer products take the rule one step at a time.
3. The rewritten change $\Delta(uv)$
  Write $\Delta(uv) = [u(x+\Delta x) - u(x)]v(x+\Delta x) + u(x)[v(x+\Delta x) - v(x)]$ and multiply it out to confirm the middle terms cancel back to the original difference.
4. Dividing by $\Delta x$ and the continuity of $v$
  Form the difference quotient, take $\Delta x \to 0$ , and record which factor needs the continuity of $v$ to become $v(x)$ .
Attempt 1 of 2