How To Find The Slope & The Equation Of The Tangent Line To The Graph At The Specified Point

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)).

Step 1

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.

Step 2

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.

Step 3

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.