Problem solving and so frustrating

[Steven G]

You are correct that if you insist that you can't do it, then you won't be able to do it.

 

[Bliman]

You start off with [math]\lim _{Δx->0} \dfrac{f(x+Δx)g(x+Δx) - f(x)g(x)}{Δx}[/math].

and you ultimately want to get

[math]\lim _{Δx->0} \dfrac{(f(x+Δx)-f(x))g(x) + f(x)(g(x+Δx)-g(x))}{Δx}[/math]
Fill in the gap!

Sorry for the late reaction.
I first tried to unravel [math]\lim _{Δx->0} \dfrac{f(x+Δx)g(x+Δx) - f(x)g(x)}{Δx}[/math]. But only the numerator.
I did that to get more insight in it. This ultimately gave me f(Δx)(g(x)+g(Δx))+f(x)g(Δx)
Then I deconstructed
[math]\lim _{Δx->0} \dfrac{(f(x+Δx)-f(x))g(x) + f(x)(g(x+Δx)-g(x))}{Δx}[/math]To see how close it is to the first and try to solve the middle step.
This got me f(Δx)g(x)+f(x)g(Δx)
As you can see f(x)g(Δx) appears in the two of them.
And that is where I am stuck.
I have written the process here to give some insight into my thinking.
Sadly I still don't know how I can get better at all of this. Don't I get it and should keep going over the material from different Calculus books to get better at it? And does that mean that I don't get it on a deep enough level?