Derivative Proof: Step-by-Step Explanation

[tanzl]

http://i359.photobucket.com/albums/oo31/tanzl/JavaPrinting-3.jpg

Not really sure whether I stated everything clearly. Please correct me.

 

[Hurkyl]

The expression

$$\frac{h}{h+k} \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$

is, at the very least, very misleading. The limit expression introduces h as a 'local variable' valid only in the scope of the limit expression. So the h's on the right cannot denote the same variable as the h's on the left.

Many mathematical styles expressly forbid overloading symbols like this in an expression, so the expression is actually grammatically incorrect.

And given that the variables h and k hadn't appeared previously (yes, I know those symbols appeared as local variables in limit expressions -- but as I've said above, those symbols cannot be referring to the same variable), I believe you've simply made a mistake.


Later, you used the equality

$$\frac{h}{h+k} \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \rightarrow 0} \frac{h}{h+k} \frac{f(x+h) - f(x)}{h}$$

which is definitely false, because the two instances of $h / (h+k)$ are not the same expression: they involve different variables (although the same symbols). Errors like this are the reason why using the same name for different variables is strongly discouraged.