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

You need to find the instantaneous slope of a curve in order to draw its tangent line.
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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.

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