### Take A Derivative by Taking The Limit of A Difference Quotient

Vocabulary and Basics First

Before one can take a derivative by taking the limit of a difference quotient, one needs to know what a difference quotient and a limit are.

Difference is a subtract expression, as in the difference between two numbers, as in the difference between 9 and 7 is 9 - 7 or 2. The difference may be between a function at one value of x and the function of another value of x is f(x2) - f(x1). To find a difference, just subtract.

Quotient is a division expression, as in the result when two numbers are divided, as in the quotient 8 ÷ 2 is 4. To find a quotient, divide. Remember also that a division expression may be written as a fraction, as in, numerator / denominator,
12/7, or
(4 -1) / (-9-3), or
[f(x2) - f(x1)]/[x2 - x1].

A difference quotient is a fraction (a division expression) in which the numerator is a difference in function values and the denominator is a difference in x values -- JUST LIKE SLOPE!

To take a limit, think "approach." Use the sketchpad in the box to examine "approach."

 Use Dynamic Analytic Geometry to Take A Limit

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 Some Examples Derivative Proofs             d(ex)/dx= ex         d( sin(x) )/dx = cos(x)