How Do You Calculate the Slope of a Tangent Line?

To find the slope of the tangent line to the graph of a function at a point, find the derivative of the function, then plug in the x-value of the point. Completing the calculation takes just a few minutes by hand, or a calculator can be used.

1. Find the point and the function

In order to find the tangent line to a graph of a function, the function must first be known. There must also be a point given where the tangent line touches the graph. Identify the value of the x-coordinate of the point. Try sketching the graph and the tangent line to get an estimate of a reasonable value for the slope.

2. Find the derivative of the function

The slope of the tangent line depends on being able to find the derivative of the function. Write down the derivative of the function, simplifying if possible. If the derivative is difficult to do by hand, consider using a calculator or computer algebra system to find the derivative.

3. Evaluate the derivative at the appropriate value

Plug the x-value from step 1 into the derivative from step 2, and evaluate the result. The number, which comes from this function evaluation, is the slope of the tangent line. Check the answer against the estimate obtained from the graph.

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