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

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Question 515569: What is the difference quotient of f(x)=x^4
Answer by drcole(72) About Me  (Show Source):
You can put this solution on YOUR website!
The difference quotient of a function f(x) is:
%28f%28x+%2B+h%29+-+f%28x%29%29%2F%28h%29
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, f%28x%29+=+x%5E4, so
f%28x+%2B+h%29+=+%28x+%2B+h%29%5E4
(multiplying out %28x+%2B+h%29%5E4, with the intermediate steps skipped --- write out the product and distribute carefully)
Substituting into the formula for the difference quotient, we get:
%28f%28x+%2B+h%29+-+f%28x%29%29%2F%28h%29
(substituting)
%284%28x%5E3%29h+%2B+6%28x%5E2%29%28h%5E2%29+%2B+4x%28h%5E3%29+%2B+h%5E4%29%2F%28h%29 (combining like terms)
4+x%5E3+%2B+6%28x%5E2%29h+%2B+4x%28h%5E2%29+%2B+h%5E3+ (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 4x%5E3 as the derivative of x%5E4.