Define differentiability of a function at a point. - Vidyadip

Question

Define differentiability of a function at a point.

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Answer

Let f(x) be a real valued function defined on an open interval (a, b) and let $\text{c}\in(\text{a, b}).$
Then f(x) is said to be differentiable or derivable at x = c iff $\lim_\limits{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}$ exists finitely.
or, $\text{f}'(\text{c})=\lim_\limits{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}.$

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