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Definitions

The derivative of a function $$f$$ at a point $$x_0$$ is the limit $$$ \frac{df}{dx}(x_0) = \lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}. $$$ If the limit exists, we say that the function is differentiable at $$x_0$$. The derivative is denoted $$\displaystyle \frac{df}{dx}(x_0)$$, $$f'(x_0)$$ or $$\dot{f}(x_0)$$.

The function $$x\mapsto f'(x)$$ is called the derivative of $$f$$.

The quantity $$\displaystyle \frac{f(x)-f(x_0)}{x-x_0}$$ is the finite difference or increment ratio of the function $$f$$ over $$[x_0,x]$$. It is the slope of the line joining the points of the graph $$\big(x_0,f(x_0)\big)$$ and $$\big(x,f(x)\big)$$.

In the limit $$x\rightarrow x_0$$ this line becomes tangential at $$x_0$$ to the graph of $$f$$. Thus $$f'(x_0)$$ is the slope of the graph at $$x_0$$.

Derivative and finite difference
Derivative and finite difference

The tangent to the graph of $$f(x)$$ at $$x_0$$ is the line with slope $$f'(x_0)$$ passing through $$\big(x_0,f(x_0)\big)$$. Its equation is $$$ y = f(x_0)+f'(x_0)(x-x_0).$$$ The tangent is the linear function that best approximates $$f$$ near $$x_0$$.

The direction of the tangent is $$\tvec = \big(1,f'(x_0)\big)$$.

The normal to the graph is the normal to the tangent. Recall that the normal is the line perpendicular to a graph. The normal is the vector $$\nvec = \big(-f'(x_0),1\big)$$. This is because $$ \tvec\cdot\nvec = 0$$.

The equation of the normal to the graph at $$x_0$$ is given by the equation $$$ f'(x_0)\big(y - f(x_0)\big) + (x-x_0) = 0.$$$

Tangent and normal
Tangent and normal