Derivative as a Rate Measure, Tangents and Normals
3.0 Equation of Tangent
3.0 Equation of Tangent
As we know that the equation of line passing through point $P({x_1},{y_1})$ and slope $m$ is written as $$y - {y_1} = m(x - {x_1}){\kern 1pt} \quad ...(1)$$ Therefore, slope ($m$) of a line at any point $P({x_1},{y_1})$ is represented by $${\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}}$$ and written as $$m = {\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}} = \tan \theta $$
Therefore, equation $(1)$ becomes $$y - {y_1} = {\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}}(x - {x_1})\quad ...(2)$$ where ${\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}}$ is the slope of the line.
We conclude that the equation of tangent at any point $P({x_1},{y_1})$ is given by $$\frac{{y - {y_1}}}{{x - {x_1}}} = {\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}}$$
Conditions:
1. When slope is $0$: To find the equation of tangent at any point $P({x_1},{y_1})$ when the slope of tangent is $0$, we put ${\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}} = 0$ in equation $(2)$, we get $$\begin{equation} \begin{aligned} y - {y_1} = 0(x - {x_1}) \\ y - {y_1} = 0 \\ y = {y_1} \\\end{aligned} \end{equation} $$ which is the equation of line parallel to $X - axis$ and passing through a point $(0,{y_1})$.
2. When slope is $\infty $: To find the equation of tangent at any point $P({x_1},{y_1})$ when the slope of tangent is $\infty $, we put ${\left( {\frac{{dy}}{{dx}}} \right)_{({x_1},{y_1})}} = \infty $ in equation $(2)$, $$\begin{equation} \begin{aligned} y - {y_1} = \infty (x - {x_1}) \\ y - {y_1} = \frac{{x - {x_1}}}{0} \\ 0\left( {y - {y_1}} \right) = x - {x_1} \\ x - {x_1} = 0 \\ x = {x_1} \\\end{aligned} \end{equation} $$
which is the equation of line parallel to $Y - axis$ and passing through a point $({x_1},0)$.
