Definition of derivative. Geometric and physical meaning

Derivative $f'(x_0)$ as a limit of secants

Average rate of change on $[x_0;x_0+\Delta x]$ (secant): $\dfrac{\Delta y}{\Delta x}=\dfrac{f(x_0+\Delta x)-f(x_0)}{\Delta x}.$ The derivative (if the limit exists and is finite): $f'(x_0)=\displaystyle\lim_{\Delta x\to0}\dfrac{f(x_0+\Delta x)-f(x_0)}{\Delta x}.$ Geometry: $f'(x_0)$ is the slope of the tangent to $y=f(x)$ at $(x_0;f(x_0))$ . Physics: if $x$ is time and $f(x)$ is position, then $f'(x_0)$ is instantaneous velocity at $x_0$ .

The derivative may fail to exist: corner, vertical tangent, discontinuity

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