SOLUTION: x^3+2x^2-x-2 Finding f'(x) from the definition of a derivative using f(x+h)-f(x)/h LIMIT -> 0. Thank you.

Algebra ->  Functions -> SOLUTION: x^3+2x^2-x-2 Finding f'(x) from the definition of a derivative using f(x+h)-f(x)/h LIMIT -> 0. Thank you.      Log On


   



Question 1073697: x^3+2x^2-x-2
Finding f'(x) from the definition of a derivative using f(x+h)-f(x)/h LIMIT -> 0.

Thank you.

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
We calculate f(x+h), then f(x+h)-f(x), then %28f%28x%2Bh%29-f%28x%29%29%2Fh,
then matrix%282%2C2%2C+lim%2C%28f%28x%2Bh%29-f%28x%29%29%2Fh%2C%22h-%3E0%22%2C%22%22%29

%22f%28x%29%22%22%22=%22%22x%5E3%2B2x%5E2-x-2

We find f(x+h):

%22f%28x%2Bh%29%22%22%22=%22%22%28x%2Bh%29%5E3%2B2%28x%2Bh%29%5E2-%28x%2Bh%29-2

%22f%28x%2Bh%29%22%22%22=%22%22%28x%5E3%2B3x%5E2h%2B3xh%5E2%2Bh%5E3%29%2B2%28x%5E2%2B2hx%2Bh%5E2%29-x-h-2

%22f%28x%2Bh%29%22%22%22=%22%22x%5E3%2B3x%5E2h%2B3xh%5E2%2Bh%5E3%2B2x%5E2%2B4hx%2B2h%5E2-x-h-2

We find f(x+h)-f(x)

%22f%28x%2Bh%29%22-%22f%28x%29%22%22%22=%22%22

%22f%28x%2Bh%29%22-%22f%28x%29%22%22%22=%22%22x%5E3%2B3x%5E2h%2B3xh%5E2%2Bh%5E3%2B2x%5E2%2B4hx%2B2h%5E2-x-h-2-x%5E3-2x%5E2%2Bx%2B2

We simplify and find that all terms not containing h cancel:

%22f%28x%2Bh%29%22-%22f%28x%29%22%22%22=%22%223x%5E2h%2B3xh%5E2%2Bh%5E3%2B4hx%2B2h%5E2-h

We factor out h

%22f%28x%2Bh%29%22-%22f%28x%29%22%22%22=%22%22h%283x%5E2%2B3xh%2Bh%5E2%2B4x%2B2h-1%29

We find %28f%28x%2Bh%29-f%28x%29%29%2Fh

%28%22f%28x%2Bh%29%22-%22f%28x%29%22%29%2Fh%22%22=%22%22%28h%283x%5E2%2B3xh%2Bh%5E2%2B4x%2B2h-1%29%29%2Fh

We cancel the h's:

%28%22f%28x%2Bh%29%22-%22f%28x%29%22%29%2Fh%22%22=%22%223x%5E2%2B3xh%2Bh%5E2%2B4x%2B2h-1

%22f%27%28x%29%22%22%22=%22%22matrix%282%2C2%2C+lim%2C%28f%28x%2Bh%29-f%28x%29%29%2Fh%2C%22h-%3E0%22%2C%22%22%29%22%22=%22%22matrix%282%2C2%2C+lim%2C%283x%5E2%2B3xh%2Bh%5E2%2B4x%2B2h-1%29%2C%22h-%3E0%22%2C%22%22%29

%22f%27%28x%29%22%22%22=%22%223x%5E2%2B3x%2A0%2B0%5E2%2B4x%2B2%280%29-1

%22f%27%28x%29%22%22%22=%22%22%22%22=%22%223x%5E2%2B4x-1

Edwin