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∞

Differentiating 1/x Straight From the Definition

Every difference quotient starts life as 0/0 — which cancellation rescues 1/x and lets the limit finally be taken?

Loading…

←Previous Secant Lines and the Limit Definition of the Derivative Next Tangents to 1/x and the Triangle of Area 2 →

Your summary note

1. The $0/0$ pitfall and the cancellation
  Write the difference quotient for $f(x)=1/x$ at $x_0$ , note that setting $\Delta x=0$ immediately gives $0/0$ , then combine over the common denominator $(x_0+\Delta x)x_0$ and factor $1/\Delta x$ out front.
2. The result $f'(x_0)=-1/x_0^2$
  Carry the numerator to $-\Delta x$ , cancel the $\Delta x$ against the factor out front to reach $-1/[(x_0+\Delta x)x_0]$ , then set $\Delta x=0$ to get $-1/x_0^2$ .
3. Graph sanity-check on the hyperbola
  Record that $f'(x_0)$ is negative everywhere matching the downward-sloping tangent, and that its magnitude shrinks as $x_0\to\infty$ matching the flattening curve.
Attempt 1 of 2