How do we derive the difference quotient?
In the beginning of the year, we learned the difference quotient song. It goes a little something like this: "f of x plus h, minus f of x, divided by the letter h, that's the difference quotient." Here's a visual:
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| http://images.tutorvista.com/cms/images/39/difference-quotient-formula.png |
Even though we've used this formula so often, we never knew where it came from. As out year reaches an end, we finally learn where this formula comes from!
If our graph has a secant line, the two points it hits will be (x, f(x)) and ((x+h), f(x+h)). h can also be represented as delta x, or the change in x. The smaller the delta x, the more the secant and tangent lines resemble one another.
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| http://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Secant-calculus.svg/250px-Secant-calculus.svg.png |
First, we know the slope formula is :
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| http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg |
If we were to plug the y and x values into the slope formula, we would get:
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| http://www.homeschoolmath.net/worksheets/equation_editor.php |
Then, since the change of x is so small, almost 0, we can ignore it in the denominator. Therefore, our equation simplifies to:
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| http://www.homeschoolmath.net/worksheets/equation_editor.php |
The smaller the delta x, or h, the more similar the secant and tangent lines become. Therefore, if we take the limit as h approaches 0, we will be able to find the slope of the tangent line. Moreover, evaluating the limit of the difference quotient is also called the derivative.
