SOLUTION: What is the difference quotient of f(x)=x^4

Question 515569: What is the difference quotient of f(x)=x^4
Answer by drcole(72) (Show Source): You can put this solution on YOUR website!
The difference quotient of a function f(x) is:

It measures the slope of the line that passes through the graph y = f(x) at x and x + h, where h is typically a very small number. If we are able to take the limit as h approaches 0, the resulting number is the derivative f'(x).
In this case, , so

(multiplying out , with the intermediate steps skipped --- write out the product and distribute carefully)
Substituting into the formula for the difference quotient, we get:

(substituting)
(combining like terms)
(canceling h)
Notice that as h goes to 0, all of the terms of the difference quotient go to 0 except for the first, leaving as the derivative of .
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