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The Difference Quotient
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f(x+h) = 3(x+h)2 = 3(x2 + 2xh + h2) = 3x2 + 6xh + 3h2
You will start with a function f(x). For example, let f(x) = 3x2 First evaluate f(x+h) f(x+h) = 3(x+h)2 = 3(x2 + 2xh + h2) = 3x2 + 6xh + 3h2
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Then subtract the original function from the expression you obtained
Then subtract the original function from the expression you obtained . You will want to use parentheses because you may run into trouble without them. 3x2 + 6xh + 3h2 – (3x2)
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Now simplify your new expression
Now simplify your new expression. You will notice that something interesting happens. 3x2 + 6xh + 3h2 – (3x2) Simplifies to 6xh + 3h2
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Now divide the expression by h.
6xh + 3h2 h
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Simplify this expression by factoring an h from the numerator and cancelling it with the h in the denominator. 6xh + 3h2 = h h(6x + 3h)
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Drum roll please Your final answer is 6x + 3h
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Let’s try another. f(x) = x2 + 5x f(x+h) = (x+h)2 + 5(x+h) = x2 + 2xh + h2 + 5x + 5h Now subtract f(x) from the above expression x2 + 2xh + h2 + 5x + 5h – (x2 + 5x) = 2xh + h2 + 5h
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Now divide by h 2xh + h2 + 5h = h h(2x + h + 5) = 2x h
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