Equation of the tangent to a curve

Reminder: derivative and tangent

The derivative f'(a) represents the slope of the tangent line to the curve f at the point with abscissa a. The equation of this tangent line is:

y = f'(a) * (x - a) + f(a)

Method

1. Calculate f(a), the y-coordinate of the point of contact.
2. Calculate f'(x) and then f'(a), the slope.
3. Write the equation by substituting into the formula.

Example

Let f(x) = x² and a = 3.

• f(3) = 9
• f'(x) = 2x, so f'(3) = 6

The equation of the tangent line is: y = 6(x - 3) + 9 = 6x - 18 + 9 = 6x - 9.

Special case: horizontal tangent

When f'(a) = 0, the tangent is horizontal: its equation reduces to y = f(a). This is exactly what happens at points where f has a local extremum.

Pitfalls

Two common mistakes:

• confusing f'(a), which is a number (the slope), with f'(x), which is a function;
• omitting the term + f(a) from the formula, which results in a line with the correct slope but which does not pass through the point on the curve.