The normal to f\left(x\right) at x=a is the line that passes through \left(a,f\left(a\right)\right) and is perpendicular to the tangent:

\begin{align}m_{\text{n}}\cdot m_{\text{t}}=-1\Leftrightarrow m_{\text{n}} & =-\frac{1}{m_{\text{t}}}\quad🚫\\ \placeholder{} & \placeholder{}\\ \placeholder{} & =-\frac{1}{f^{\prime}\left(a\right)}\quad🚫\end{align}

Using point slope form the equation of the tangent is:

\begin{align}y-f\left(a\right) & =m_{n}\cdot\left(x-a\right)\quad🚫\\ \placeholder{} & \placeholder{}\\ \Rightarrow y & =m_{n}x-m_{n}a+f\left(a\right)\quad🚫\end{align}

Example

Find the normal to f\left(x\right)=x-2x^2 at x=2.

\begin{align}f^{\prime}\left(x\right)=1-4x\Rightarrow m_{t} & =f^{\prime}\left(2\right)=-7\\ \Rightarrow m_{n} & =\frac17\end{align}

The equation is:

\begin{align}y-f\left(2\right) & =\frac17x-\frac17\cdot2\\ \placeholder{} & \placeholder{}\\ y & =\frac17x-\frac27+\left(2-2\cdot2^2\right)\\ \placeholder{} & \placeholder{}\\ y & =\frac{x}{7}-\frac{44}{7}\end{align}

Differentiation

Tangents and normals