Finding the Equation of a Tangent Line
Using the First Derivative
Certain problems in Calculus I call for using the first derivative to find the equation of the tangent line to a curve at a specific point.
The following diagram illustrates these problems.

There are certain things you must remember from College Algebra (or similar classes) when solving for the equation of a tangent line.
Recall :
• A Tangent Line is a line which locally touches a curve at one and only one point.
•

The slope-intercept formula for a line is y = mx + b, where m is the slope of the line and b is the y-intercept.

•

The point-slope formula for a line is y – y1 = m (x – x1). This formula uses a point on the line, denoted by (x1, y1), and the slope of the line, denoted by m, to calculate the slope-intercept formula for the line.

Also, there is some information from Calculus you must use:
Recall:
• The first derivative is an equation for the slope of a tangent line to a curve at an indicated point.
• The first derivative may be found using:

A) The definition of a derivative :

lim h →0

f (x + h ) − f ( x ) h B) Methods already known to you for derivation, such as:
• Power Rule
• Product Rule
• Quotient Rule
• Chain Rule
(For a complete list and description of these rules see your text)

With these formulas and definitions in mind you can find the equation of a tangent line.
Consider the following problem:
Find the equation of the line tangent to f ( x ) = x 2 at x = 2.
Having a graph is helpful when trying to visualize the tangent line. Therefore, consider the following graph of the problem:
8
6
4
2

-3

-2

-1

1

2

3

The equation for the slope of the tangent line to f(x) = x2 is f '(x), the derivative of f(x).
Using the power rule yields the following: f(x) = x2 f '(x) = 2x

(1)

Therefore, at x = 2, the slope