how to solve using limit definition of the derivative for f(x)=2x^3+5x^2-7x-4

my ans:

6x^2+10x-7

plse correct me...

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- Dec 10th 2008, 08:59 PMmathseekusing limit derivative?
how to solve using limit definition of the derivative for f(x)=2x^3+5x^2-7x-4

my ans:

6x^2+10x-7

plse correct me... - Dec 10th 2008, 09:18 PMProve It
- Dec 10th 2008, 09:23 PMmathseeklimit def
than what is limit definition??

- Dec 10th 2008, 09:27 PMo_O
Definition of a derivative: $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

- Dec 10th 2008, 09:32 PMmathseeklimit def
so when i answer this question above....i just give my ans will do right?

- Dec 10th 2008, 09:40 PMProve It
No - I thought you had gone through the rigmarole of using the limit definition and gotten that answer.

$\displaystyle f(x) = 2x^3 + 5x^2 -7x - 4$

$\displaystyle f(x + h) = 2(x + h)^3 + 5(x + h)^2 - 7(x + h) - 4$

$\displaystyle = 2(x^3 + 3x^2h + 3xh^2 + h^3) + 5(x^2 + 2xh + h^2) - 7x -7h -4$

$\displaystyle = 2x^3 + 6x^2h + 6xh^2 + 2h^3 + 5x^2 + 10xh + 5h^2 - 7x - 7h - 4$

So $\displaystyle f(x + h) - f(x) = 6x^2h + 6xh^2 + 2h^3 + 10xh + 5h^2 - 7h$

$\displaystyle = h[6x^2 + 6xh + 2h^2 + 10x + 5h - 7]$.

$\displaystyle \frac{f(x+h) - f(x)}{h} = 6x^2 + 6xh + 2h^2 + 10x + 5h - 7$

$\displaystyle \lim_{h \to 0}\frac{f(x+h) - f(x)}{h} = 6x^2 + 6x(0) + 2(0)^2 + 10x + 5(0) - 7$

$\displaystyle = 6x^2 + 10x - 7$. - Dec 10th 2008, 09:42 PMo_O
Well you have to show your work probably and you do so by going through the grunt work of expanding and simplifying in order to evaluate the limit:

$\displaystyle \begin{aligned}f'(x) & = \lim_{h \to 0} \frac{2(x+h)^3+5(x+h)^2-7(x+h) - 4 \ - \ \left(2x^3 + 5x^2 - 7x - 4\right)}{h} \\ & = \lim_{h \to 0} \frac{2(x^3 + 2x^2h + 2xh^2 + h^3) +5(x^2 + 2xh + h^2) - \cdots}{h} \end{aligned}$

Eventually you can factor out an h from the top to cancel it with the bottom and you should get your answer. Like I said, a lot of grunt work.