SOLUTION: (f(x+h)-f(x))/(h) Let f(x) = -2x^(2)- x + 3 h ≠ 0 Find and simply the difference quotient

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Question 1202783: (f(x+h)-f(x))/(h) Let f(x) = -2x^(2)- x + 3
h ≠ 0
Find and simply the difference quotient

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: -4x-2h-1


Work Shown:

We first compute f(x+h)

f%28x%29+=+-2x%5E2-x%2B3

f%28x%2Bh%29+=+-2%28x%2Bh%29%5E2-%28x%2Bh%29%2B3 Each x replaced with (x+h).

f%28x%2Bh%29+=+-2%28x%2Bh%29%28x%2Bh%29-x-h%2B3

f%28x%2Bh%29+=+-2%28x%5E2%2B2xh%2Bh%5E2%29-x-h%2B3 FOIL rule

f%28x%2Bh%29+=+-2x%5E2-4xh-2h%5E2-x-h%2B3

Now subtract off f(x)

f%28x%2Bh%29-f%28x%29+=+%28-2x%5E2-4xh-2h%5E2-x-h%2B3%29+-+%28-2x%5E2-x%2B3%29

f%28x%2Bh%29-f%28x%29+=+-2x%5E2-4xh-2h%5E2-x-h%2B3+%2B+2x%5E2%2Bx-3

f%28x%2Bh%29-f%28x%29+=+-4xh-2h%5E2-h

f%28x%2Bh%29-f%28x%29+=+h%28-4x-2h-1%29

The last step is to divide over h.

%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%28-4x-2h-1%29%29%2Fh



%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28cross%28h%29%28-4x-2h-1%29%29%2F%28cross%28h%29%29

%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+-4x-2h-1

Note: As h approaches 0, then -4x-2h-1 approaches -4x-1 which is the derivative of f%28x%29+=+-2x%5E2-x%2B3.