How do you find the equation of a tangent to a graph?

The three steps

First, complete the point. A question usually gives only the x-coordinate, and the y-coordinate comes from substituting it into the function — into the function, not the derivative.

Second, find the slope. Differentiate, then substitute the same x into the derivative. The number that comes out is the slope of the tangent at that point.

Third, write the line. With a point and a slope, the equation of a straight line is immediate, and there is no further calculus involved.

Completing the point

Substituting into the wrong one of the two functions is the commonest error here, and it produces a line that is not tangent to anything.

A quick check: the point must lie on the graph. If you substitute its coordinates back into the original function and the two sides do not agree, the point is wrong before the slope is even considered.

When the slope is given instead of the point

Some questions give a slope and ask where the tangent has it. That reverses the work: set the derivative equal to that slope and solve for x.

There may be more than one answer, since a derivative can take the same value at several points. Find all of them, then complete each into a point and write a line for each.

Worked examples

1. Find the tangent to f(x) = x squared at x = 3

   1. Complete the point: f(3) is 9, so the point is (3, 9)
   2. Differentiate: 2x
   3. Substitute: the slope is 6
   4. Write the line through (3, 9) with slope 6

   Answer: y = 6x − 9

2. Find the tangent to f(x) = x squared minus 4x at x = 1

   1. Complete the point: 1 minus 4 is minus 3, so (1, −3)
   2. Differentiate: 2x minus 4
   3. Substitute: 2 minus 4, so the slope is minus 2
   4. Write the line through (1, −3) with slope minus 2

   Answer: y = −2x − 1

3. Where does f(x) = x squared have a tangent of slope 10?

   1. The slope is given, so set the derivative equal to it
   2. 2x equals 10
   3. Solve: x equals 5
   4. Complete the point: f(5) is 25

   Answer: At the point (5, 25)

Common mistakes

Substituting into the derivative to get the y-coordinate

The point lies on the graph of the function, so its y-coordinate comes from the function. The derivative supplies the slope and nothing else.

Writing the line with the x-coordinate as the intercept

A point and a slope give the line through that point, not a line crossing the axis there. Substituting the point into the line equation and solving for the intercept avoids it.

Stopping at one answer when the slope was given

A derivative can take the same value at several points, so setting it equal to a slope can have more than one solution. Each solution is a separate tangent.

More in Introduction to Derivatives

• What is an average rate of change and how does it relate to the slope of a chord?
• What is the derivative of a function at a point?
• How do you differentiate a polynomial?
• When do you use the product, quotient and chain rules?
• How do you find intervals of increase and decrease and points of extremum?