Example Find the tangent line of the following curve at the point corresponding to $x = a$.

$$\displaystyle{f(x) = x^{2} - 5x + 3, \ a = 2}$$

Answer The slope of the tangent line at $x = a$ is given by $f'(a)$. So,

$$f'(2) = 2*2 - 5 = -1$$

We know every straight line can be determined by knowing the slope $m$ and the point $(a,f(a))$ goes through. Then the equation of the tangent line is given by $y = m(x-a) + f(a)$．Now we find the point where the tangent line goes through.

$$f(2) = 4 -10+3 = -3$$

Thus the equation of the tangent line is

$$y = -(x-2) -3 = -x -1$$

To find the normal line, it is enough to know that the slope $m$ of the tangent line and the slope $m'$ of the normal line, $mm' = -1$. To see this, consider the vector $(1,m)$ on the tangent line and the vector $(1,m')$ on the normal line. Then these vector are orthogonal which implies $(1,m)\cdot (1,m') = 1 + mm' = 0$. Thus，$mm' = -1$. Therefore, the equation of normal line is

$$y = (x-2)-3 = x-5$$