12.5 Derivatives: Introduction and First Principles

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  1. Definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
  2. Substitute: f(x+h)=(x+h)2=x2+2xh+h2f(x+h) = (x+h)^2 = x^2 + 2xh + h^2.
  3. Difference: f(x+h)−f(x)=(x2+2xh+h2)−x2=2xh+h2=h(2x+h)f(x+h) - f(x) = (x^2 + 2xh + h^2) - x^2 = 2xh + h^2 = h(2x+h).
  4. Divide by h: f(x+h)−f(x)h=2x+h\frac{f(x+h)-f(x)}{h} = 2x + h.
  5. Limit: lim⁡h→0(2x+h)=2x\lim_{h \to 0} (2x + h) = 2x. Therefore, ddx(x2)=2x\frac{d}{dx}(x^2) = 2x.

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