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I'm having trouble with proving that the derivative of f(x)*g(x) is f'(x)*g(x)+f(x)*g'(x).

Now, I've already seen the actual proof, and I can understand its reasoning, but the first time I tried to prove without looking at the solution, this is what I wrote before I became rather confused:

So, using the limit definition of the derivative, I get that the derivative of f(x)*g(x) is:

[itex]

\displaystyle\lim_{\Delta x \rightarrow 0} {\frac{f(x+\Delta x)g(x+\Delta x) - f(x)g(x)} {\Delta x}}

[/itex]

I can rewrite this as:

[itex]

\displaystyle\lim_{\Delta x \rightarrow 0} {(f(x+\Delta x)\frac{g(x+\Delta x) }{\Delta x} - f(x)\frac{g(x)}{\Delta x})}

[/itex]

Then I used some limit rules (multiplication and subtraction of limits) to rewrite the limit:

[itex]

\displaystyle\lim_{\Delta x \rightarrow 0} {f(x+\Delta x)} \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x+\Delta x)}{\Delta x}} - \displaystyle\lim_{\Delta x \rightarrow 0} {f(x)} \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x)}{\Delta x}}

[/itex]

I believe that, because f(x) and g(x) are differentiable (and therefore continuous), this evaluates to:

[itex]

f(x) \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x+\Delta x)}{\Delta x}} - f(x) \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x)}{\Delta x}}

[/itex]

Then I can factor out f(x) and be left with :

[itex]

f(x)( \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x+\Delta x)}{\Delta x}} - \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x)}{\Delta x}})

[/itex]

Then, because the the limit of a difference is the same as the difference of the two terms' limits:

[itex]

f(x)( \displaystyle\lim_{\Delta x \rightarrow 0}{\frac{g(x+\Delta x)-g(x)}{\Delta x}})

[/itex]

I finally get: [itex]

f(x)g'(x)

[/itex]

I realize that this isn't the right answer. Where am I going wrong?

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