Differentiation and integration » Tangent line to graph  

In the example below, the explanation is given using an example.

Given is function f (x) = 0.5x³ – 4x + 3.
On the graph of f lies point P with x = –2

Use the derivative to make the formula for tangent line k : y = ax + b in point P.

First calculate the derivative.
f ' (x) = 1.5x² – 4.

Now calculate a. This is the gradient of point P.
a = 1.5(–2)² – 4 = 2

Now calculate the y-coordinate of point P.
f (–2) = 0.5(–2)³ – 4 · (–2) + 3 = 7

Fill this in into the formula of line k : y = ax + b so you can solve this and calculate b.
 7 = 2 · (–2) + b
 7 = –4 + b
 b = 11

The formula for the tangent line is k : y = 2x + 11.