11TH•Mathematics•Differentiation•13.2 Differentiation from First Principles

13.3 Notation and First Principles

Derivative notation, four-step definition method, and the power rule for integer n.

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With $y = f(x)$ and $\delta y = f(x+\delta x) - f(x)$ :

\dfrac{dy}{dx} = \lim_{\delta x \to 0} \dfrac{\delta y}{\delta x} = f'(x)

$\dfrac{dy}{dx}$ is a single symbol (not a quotient). Other notations: Leibniz $\dfrac{dy}{dx}$ , Lagrange $f'(x)$ , Euler $Df(x)$ .

Alternate form at $x = a$ :

f'(a) = \lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}

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