The derivative of a continuous function at x is the slope of the tangent line to the curve at x. The attached pdf file develops the idea of a derivative first using slopes of secant lines and then introducing and explaining the difference quotient in detail. An example and an explanation are provided for using the limit of the difference quotient to find the derivative of the function f(x) = x^2 + 4x.
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The attached pdf document provides a detailed explanation of the definition of the derivative. Commonly used notation is introduced. An example is given illustrating how to apply the definition of the derivative to finding the derivative of a function.