Here is a good example
\begin{aligned}\text {Differentiation of}\,x^2+3xy+4y^2=58\\
2x+3y+3x\frac{dy}{dx}+8y{dy}{dx}&=0\\
(3x+8y)\frac{dy}{dx}+2x+3y&=0\\
(3x+8y)\frac{dy}{dx}&=-(2x+3y)\\
\frac{dy}{dx}&=-\frac{(2x+3y)}{(3x+8y)}\\
\text{at the point}\,\,(2,\,3)\,x=2,\,y=3\quad \frac{dy}{dx}&=-\frac{(4+9)}{6+24}\\
&=-\frac{13}{30}\\
\text{if gradient is}\,m\,\text{gradient of normal is}\,&=\frac{-1}{m}\\
\frac{-1}{\frac{-13}{30}}=-\frac{1}{1}\div-\frac{13}{30}&=-\frac{1}{1}\times-\fr ac{30}{13}=\frac{30}{13}\\
\text{For equation if line through}\,\,(2,\,3)\\
\frac{y-3}{x-2}&=\frac{30}{13}\\
13y-39&=30x-60\\
30x-13y-21&=0
\end{aligned}