Derivative f′(x0)f'(x_0) as a limit of secants
Average rate of change on [x0;x0+Δx][x_0;x_0+\Delta x] (secant): ΔyΔx=f(x0+Δx)−f(x0)Δx.\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′(x0)=lim⁡Δx→0f(x0+Δx)−f(x0)Δx.f'(x_0)=\displaystyle\lim_{\Delta x\to0}\dfrac{f(x_0+\Delta x)-f(x_0)}{\Delta x}. Geometry: f′(x0)f'(x_0) is the slope of the tangent to y=f(x)y=f(x) at (x0;f(x0))(x_0;f(x_0)). Physics: if xx is time and f(x)f(x) is position, then f′(x0)f'(x_0) is instantaneous velocity at x0x_0.
The derivative may fail to exist: corner, vertical tangent, discontinuity
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