Definition of Derivative

dod

Let f(x) contains an open interval containing x_0 in its domain, the function is differentiable if the limit


L = \lim\limits_{\Delta x \to0} \dfrac{f(x_0+\Delta x )-f(x)}{\Delta x }


exist.

f'(x_0) = \lim\limits_{\Delta x \to0} \dfrac{f(x_0+\Delta x )-f(x_0)}{\Delta x } = \lim\limits_{\Delta x \to0}\dfrac{\Delta y}{\Delta x } = \lim\limits_{ x \to x_0} \dfrac{f(x )-f(x_0)}{x - x_0 } = \lim\limits_{ h \to0} \dfrac{f(x_0+h )-f(x_0)}{h }= \lim\limits_{\Delta x \to0} \dfrac{f(x+\Delta x )-f(x)}{\Delta x }

or

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