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 pointslope 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 pointslope formula at the specified point. For example, at point (2, 16) with slope = 24, the pointslope 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 yintercept to be at 32 with a very steep slope reasonably equating to 24.