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# 2: Derivatives

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The derivative of a function $$f$$ is another function, $$f'$$, defined as $f'(x) \;\equiv\; \frac{df}{dx} \;\equiv\; \lim_{\delta x \rightarrow 0} \, \frac{f(x + \delta x) - f(x)}{\delta x}.$ This kind of expression is called a limit expression because it involves a limit (in this case, the limit where $$\delta x$$ goes to zero).

If the derivative exists within some domain of $$x$$ (i.e., the above limit expression is mathematically well-defined), then we say $$f$$ is differentiable in that domain. It can be shown that a differentiable function is automatically continuous.

Graphically, the derivative represents the slope of the graph of $$f(x)$$, as shown below: Figure $$\PageIndex{1}$$

If $$f$$ is differentiable, we can define its second-order derivative $$f''$$ as the derivative of $$f'$$. Third-order and higher-order derivatives are defined similarly.

2: Derivatives is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.