Tangent as the graph of a linear function
If ff is differentiable at x0x_0, the tangent line to y=f(x)y=f(x) at (x0;f(x0))(x_0;f(x_0)) is y=f(x0)+f′(x0)(x−x0).y=f(x_0)+f'(x_0)(x-x_0). f′(x0)f'(x_0) is the slope; the line y=kx+by=kx+b with k=f′(x0)k=f'(x_0), b=f(x0)−f′(x0)x0b=f(x_0)-f'(x_0)x_0. The normal (perpendicular to the tangent at the same point) when f′(x0)≠0f'(x_0)\neq0 has slope −1/f′(x0)-1/f'(x_0).
If f′(x0)=0f'(x_0)=0 — the tangent is horizontal (y=f(x0)y=f(x_0))
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