How to Find the Equation of a Tangent Line of a Function

The equation for an arbitrary tangent to the graph of a function is given by the point-slope equation:

Formula

y - y_{1} = f^{'} (x_{1}) (x - x_{1})

where $(x_{1}, y_{1})$ is the point of tangency and $f^{'} (x_{1})$ is the slope. After substituting into the formula, you always solve for $y$ .

When solving differential equations, you are often asked to find the tangent to a graph at a certain point. It is then useful to use the fact that

y^{'}

from the differential equation can be used to find

f^{'} (x_{1})

. You do as follows:

Rule

1.: Solve the differential equation with respect to $y^{'}$ .
2.: Insert the $x_{1}$ -value at the point of intersection with the tangent, and find the value of $y^{'} (x_{1})$ .
3.: Enter $x_{1}$ , $y_{1}$ and $y^{'} (x_{1})$ into the point-slope equation and find the equation for the tangent.