This is from betterexplained.com. It's a really cool website with lots of intuitive explanations of maths concepts. This helped me understand Pythagoras' theorem. Actually my go-to website for intuitive explanations of concepts.
Pythagoras' theorem http://betterexplained.com/wp-content/uploads/pythagorean/proof1.png
These are similar triangles. This diagram also makes something very clear:
Area (Big) = Area (Medium) + Area (Small) Makes sense, right? The smaller triangles were cut from the big one, so the areas must add up. And the kicker: because the triangles are similar, they have the same area equation.
Let's call the long side c (5), the middle side b (4), and the small side a (3). Our area equation for these triangles is:
Area = F * hypotenuse^2
where F is some area factor (6/25 or .24 in this case; the exact number doesn't matter). Now let's play with the equation:
Area (Big) = Area (Medium) + Area (Small)
F c^2 = F b^2 + F a^2
Divide by F on both sides and you get:
c^2 = b^2 + a^2
Which is our famous theorem! You knew it was true, but now you know why.
This explains the product rule:
betterexplained http://betterexplained.com/wp-content/uploads/derivatives/productrule.png