Question

What is the equation of a normal line?

Answer 1

Normal line:

The normal line is the line that is perpendicular to the tangent line at the point of tangency.

Let $y=f\left(x\right)$ is the function of a curve.

Then, the slope of the normal line at $x=x_0$ is the negative reciprocal of the derivative of the function,

The slope of the normal line $\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{f}\mathbf{\text{'}}\left(x_0\right)}$
Hence, the equation of the normal line at the point $\left(x_0,y_0\right)$ is,

$\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{f}\mathbf{\text{'}}\left(x_0\right)}\left(x-x_0\right)$

Hence,the normal line is $\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{f}\mathbf{\text{'}}\left(x_0\right)}\left(x-x_0\right)$