- #1
Lemniscates
- 9
- 0
Hello all,
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?
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?
Last edited: