How to Find the Slope & the Equation of the Tangent Line to the Graph at the Specified Point

By Luc Braybury; Updated April 24, 2017
You need to find the instantaneous slope of a curve in order to draw its tangent line.

A tangent line is a straight line that touches only one point on a given curve. In order to determine its slope it is necessary to understand the basic differentiation rules of differential calculus in order to find the derivative function f '(x) of the initial function f(x). The value of f '(x) at a given point is the slope of the tangent line at that point. Once the slope is known, finding the equation of the tangent line is a matter of using the point-slope formula: (y - y1) = (m(x - x1)).

Differentiate the function f(x) in order to find the slope of the graph at a specified point. For example, if f(x) = 2x^3, using the rules of differentiation when find f '(x) = 6x^2. To find the slope at point (2, 16), solving for f '(x) finds f '(2) = 6(2)^2 =24. Therefore, the slope of the tangent line at point (2, 16) equals 24.

Solve for the point-slope formula at the specified point. For example, at point (2, 16) with slope = 24, the point-slope equation becomes: (y - 16) = 24(x - 2) = 24x - 48; y = 24x -48 + 16 = 24x - 32.

Check your answer to make sure it makes sense. For example, graphing the function 2x^3 alongside its tangent line y = 24x - 32 finds the y-intercept to be at -32 with a very steep slope reasonably equating to 24.

About the Author

Luc Braybury began writing professionally in 2010. He specializes in science and technology writing and has published on various websites. He received his Bachelor of Science in applied physics from Armstrong Atlantic State University in Savannah, Ga.