Definition of a Derivative

I copy the definitions of three different types of derivatives from [1]:

$ \lim_{\Delta x \rightarrow 0} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big|_{a}$

$ \lim_{\Delta x \rightarrow 0^+} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big|_{a^+}$

$ \lim_{\Delta x \rightarrow 0^-} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big|_{a^-}$

These definitions are best understood geometrically in terms of secant and tangent lines. Note that $ \Delta x$ never equals zero because otherwise the fraction would be undefined. The tangent line intersects the function at only one point.

[1] David V. Widder. Advanced Calculus. Dover 1989.

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