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Definitions

Derivative and differentiability

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

Tangent and derivative

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