Introduction to Derivatives

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

Quick answer

To find the equation of a tangent to a graph, find the y-coordinate by plugging the x-value into the original function, find the slope by plugging the x-value into the derivative, and write the line equation using point-slope form y - y1 = m(x - x1). The original function yields the point, while the derivative yields the slope.

The topic explained

The three steps to find a tangent line

Finding the equation of a tangent to a graph takes three orderly steps: find the point, find the slope, and assemble the line. Calculus problems frequently provide only the x-coordinate. Your first task is to calculate the corresponding y-coordinate by substituting x into the original function f(x). You must use the original formula here, because the point of tangency sits directly on the graph of the function.

Next, determine the slope of the line by finding the derivative f'(x) and substituting the same x-value into it. The derivative calculates the slope at that single location. Finally, take your point (x1, y1) and slope m and put them into the straight line equation y - y1 = m(x - x1). Once you reach this stage, the calculus is complete.

Worked example with real numbers

Consider the function f(x) = x^2 - 4x at x = 1. First, find the y-coordinate: evaluate f(1) = 1^2 - 4(1) = -3. The point of contact is (1, -3).

Second, differentiate f(x) to find the slope formula: f'(x) = 2x - 4. Substitute x = 1 to find the slope: m = 2(1) - 4 = -2. Third, write the line equation using the point (1, -3) and slope -2: y - (-3) = -2(x - 1). Simplifying gives y + 3 = -2x + 2, which results in y = -2x - 1.

When the slope is given instead of the point

Some problems ask you to find a tangent that is parallel to a given line, or specify a slope directly. In this situation, reverse the process: set the derivative equal to the required slope and solve the equation for x.

For instance, to find where f(x) = x^2 has a tangent of slope 10, set f'(x) = 2x equal to 10, which gives x = 5. Substitute x = 5 back into the original function f(5) = 25 to establish the full point (5, 25). If solving the derivative produces multiple values for x, each value represents a distinct point with its own separate tangent line.

Free printable worksheet

Finding the Equation of a Tangent Line

Intended for 12th-grade calculus students. Use derivatives to determine slopes, identify points of contact, and write tangent line equations in slope-intercept form.

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Finding the Equation of a Tangent LineName:

Find the equation of the tangent line to f(x) = x^2 at x = 3.
Find the equation of the tangent line to f(x) = 3x^2 - 5x + 2 at x = 2.
Find the equation of the tangent line to f(x) = x^3 - 2x at x = -1.
Find the point on the graph of f(x) = x^2 + 6x where the tangent line has a slope of 4.
Find the equation of the tangent to f(x) = 2x^2 that is parallel to the line y = 8x - 3.
Find the equations of all horizontal tangent lines to the curve f(x) = x^3 - 3x.

Common questions

Why do I substitute x into the original function instead of the derivative to find y?

The tangent line touches the curve at a point belonging to the graph. Therefore, its coordinates must satisfy the original equation f(x). The derivative f'(x) provides the slope at that point, never the y-coordinate.

What does it mean when a problem says the tangent is parallel to a line?

Parallel lines share identical slopes. If a question states that the tangent is parallel to y = 4x - 7, the slope of your tangent line must be 4. Set f'(x) = 4 and solve for x to locate the point.

Can a function have more than one tangent line with the same slope?

Yes. Polynomials of degree three or higher, as well as periodic curves, frequently turn back on themselves. Setting f'(x) equal to a slope can produce multiple real solutions, each producing a distinct tangent line.

Can Lessi help me practice tangent line problems?

Yes. Lessi is an AI tutor that can guide you through derivative rules and check your line equations step by step. Your first 60 minutes of lessons are free with an account, and Ask Lessi answers specific questions using those same free minutes.

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