Calculus: find the derivative from the limit definition

[sareevan]

Use the definition of the derivative:

. . . . .\(\displaystyle \L f'(x)\,=\,\,\begin{array}{c}lim\\h\rightarrow 0\end{array}\,\,\frac{f(x\,+\,h)\,-\,f(x)}{h}\)

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Edited by stapel -- reason for edit: clarity of formatting

 

[galactus]

And do what?.

 

[sareevan]

Calculus

use algebra to evaluate the limit.

 

[stapel]

Re: Calculus

use algebra to evaluate the limit.

What limit? You haven't included any function f(x) to evaluate.

Please reply with the full and exact (word-for-word) text of the exercise, the complete instructions, and a clear listing of everything you have done so far.

For instance, you would start with listing, line-by-line, your evaluation of f at x + h, then your simplification of this evaluation, then your subtraction of f(x) from your simplification of f(x + h), and so forth. In other words, you can at least show the algebra part of this exercise. Then please specify where you are stuck on the calculus part.

Thank you.

Eliz.

 

[sareevan]

calculus

Use the definition of the derivative, f'(x) = lim f(x+h)-f(x)/x , to find the derivative of f(x) = x^2+3

 

[stapel]

Re: calculus

Use the definition of the derivative, f'(x) = lim f(x+h)-f(x)/x , to find the derivative of f(x) = x^2+3

So how far have you gotten?

Evaluating f(x + h) and f(x) is just algebra, as is finding f(x + h) - f(x). So you know how to do that part. Once you've simplified, then see if you can factor an "h" out. If so, then cancelling off the h will be simple. If not, then the limit might be a little harder to find.

Please reply showing the above work, and any futher steps you've been able to complete. Thank you.

Eliz.

 

[sareevan]

calculus

f'(x) = lim f(x+h)-f(x)/h
f(x) = x^2 +3

f'(x) = lim f(x+h) - f(x)/h = lim (x+3)(x+1)-x/h = lim (x+3)(x+1)-x/h * (x+3)(x+1)-x/(x+3)(x+1)-x = 0/h(x+3)(x+1)-x = 0/x^2+3-x

 

[stapel]

lim (x+3)(x+1)-x/h

Where is "(x + 3)(x + 1)" coming from? Isn't the function supposed to be "x² + 3"? Or is this a new exercise with a new function?

Aren't you supposed to evaluate f at x + h? Where did you do that? I'm not seeing an "h" in your work...?

Aren't you supposed to divide the entire "f(x + h) - f(x)" expression by "h"? According to what you've posted, you're only divided "x" by "h". Where is the "x" coming from, by the way?

Please try again, this time following the instructions:

1) Evaluate f(x) = x² + 3 at x + h.

2) Simplify.

3) From (2), subtract f(x).

4) Simplify.

5) Factor an "h" out of (4).

6) Divide (5) by "h".

7) Cancel off the common factor of "h".

8) Take the limit (in this case, just evaluate) at h = 0.

If you get stuck, please reply showing your work for each of these steps. Thank you.

Eliz.

 

[galactus]

No offense, but you're sure turning this relatively simple task into a monstrosity.

Try this:

\(\displaystyle \L\\\lim_{x\to\0}\frac{(x+h)^{2}+3-(x^{2}+3)}{h}\)

\(\displaystyle \L\\\lim_{x\to\0}\frac{x^{2}+2xh+h^{2}+3-x^{2}-3}{h}\)

Now, simplify and finish up?. Okey-doke?.