Difference Quotient Question

[Jason76]

Two versions of the Difference Quotient (for finding the derivative)

\(\displaystyle f'(x) = \lim x\rightarrow 0 [\dfrac{f(x + \Delta x) - f(x)}{\Delta x}]\)

vs

\(\displaystyle f'(c) = \lim x\rightarrow c[\dfrac{f(x) - f(c)}{x - c}]\)

Under which occasions do we use one vs the other?

 

[Romsek]

Two versions of the Difference Quotient (for finding the derivative)

\(\displaystyle f'(x) = \lim x\rightarrow 0 [\dfrac{f(x + \Delta x) - f(x)}{\Delta x}]\)

vs

\(\displaystyle f'(c) = \lim x\rightarrow c[\dfrac{f(x) - f(c)}{x - c}]\)

Under which occasions do we use one vs the other?

The first form is more of a general statement about f.

The second form is more of a specific statement about f at the point c.

That's really about all you can say. The two forms are entirely equivalent.

 

[pka]

Two versions of the Difference Quotient (for finding the derivative)

\(\displaystyle f'(x) = \lim x\rightarrow 0 [\dfrac{f(x + \Delta x) - f(x)}{\Delta x}]\)

vs

\(\displaystyle f'(c) = \lim x\rightarrow c[\dfrac{f(x) - f(c)}{x - c}]\)

You have the notation in #1 wrong.

\(\displaystyle f'(x) =\displaystyle \lim_{\Delta x\rightarrow 0} [\dfrac{f(x + \Delta x) - f(x)}{\Delta x}]\)

 

[Jason76]

Two versions of the Difference Quotient (for finding the derivative)

\(\displaystyle f'(x) = \lim \Delta x\rightarrow 0 [\dfrac{f(x + \Delta x) - f(x)}{\Delta x}]\) Changed notation

vs

\(\displaystyle f'(c) = \lim x\rightarrow c[\dfrac{f(x) - f(c)}{x - c}]\)