For a function ƒ defined on an open set U having the point X:(x(adsbygoogle = window.adsbygoogle || []).push({}); _{1},x_{2},...,x_{n})

and the point ||H|| such that the point X + H lies in the set we try to

define the meaning of the derivative.

[tex]\frac{f(X \ + \ H) \ - \ f(X)}{H}[/tex] is an undefined quantity, what does it mean

to divide by a vector???

The idea is to use

[tex] f'(x) \ = \ \lim_{h \to 0} \frac{f(x \ + \ h) \ - \ f(x)}{h}[/tex]

and define

[tex] g(h) \ = \ \lim_{h \to 0} \frac{f(x \ + \ h) \ - \ f(x)}{h} \ - \ f'(x)[/tex]

for h ≠ 0 but notice that [tex] \lim_{h \to 0} \ g(h) \ = \ 0 [/tex].

If we then write

[tex] g(h) \ + \ f'(x) \ = \ \frac{f(x \ + \ h) \ - \ f(x)}{h} [/tex]

[tex] g(h) \cdot h \ + \ f'(x) \cdot h \ = \ f(x \ + \ h) \ - \ f(x) [/tex]

this holds true as long as h ≠ 0 because if h = 0 the derivative term, f'(x) makes no sense.

If the above relation holds for a function whether we've chosen h to be either positive or

negative we can then claim the function differentiable seeing as the limit exists

& the relation holds true. Therefore h can be rewritten as |h| and the above as

[tex] g(h) \cdot |h| \ + \ f'(x) \cdot h \ = \ f(x \ + \ h) \ - \ f(x) [/tex]

(My book only gives the h indicated above with |h| & I'm not sure why Lang doesn't give

the h beside the f'(x) absolute value signs one too? I think it's because he divides in a

minute by h & wants to get rid of the h beside the f'(x) & leave the g(h)•h one but it seems

to me like picking & choosing things as you please...)

So, if the function is differentiable there exists a function g such that

[tex] g(h) \cdot |h| \ + \ f'(x) \cdot h \ = \ f(x \ + \ h) \ - \ f(x) \ , \ \lim_{h \to 0} \ g(h) \ = \ 0 [/tex]

so we can define the following

[tex] g(h) \cdot \frac{|h|}{h} \ + \ f'(x) \ = \ \frac{f(x \ + \ h) \ - \ f(x)}{h} [/tex]

and in the limit we find equality above as the g(h) term makes the fraction of |h|/h

dissappear.

Is all of this correct? What is with the absolute value situation? Why are we picking &

choosing our definitions with |h| when it's convenient?(Or, more accurately - what am I not getting? :tongue:)

Thanks for taking the time to read that!

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# Homework Help: Differentiability & Gradient?

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