# Proof of the Product Rule

1. Mar 16, 2012

### Dramacon

Hello there!

I understand the product rule and how/why it works, and the proof makes sense,
but just out of curiosity, why would it be incorrect on a mathematical/application level to say that d/dx f(x)g(x) = f'(x)g'(x)

I know that it's wrong, and the product rule is the one that we're supposed to use, but I don't understand on a practical level WHY it's wrong.

2. Mar 16, 2012

### Staff: Mentor

Because it doesn't give the right answer!

Here's an example: let f(x) = x2, and g(x) = x2. Then by your version of the product rule, d/dx(f(x) g(x)) = 2x * 3x2 = 6x3.

OTOH, f(x)*g(x) = x5, so d/dx(f(x)*g(x) ) = 5x4.

The real product rule gives this result, which you can confirm.

3. Mar 16, 2012

### emailanmol

Well, product rule is something which can be derived in few steps.

Let y= f(x) and z= g(x)
Let h(x)=f(x)g(x)
For any x, h(x)=f(x)g(x)=yz
To map changes, we say that if x increases by dx, let change in y be dy and change in z be dz.
So h(x+dx)= f(x+dx)g(x+dx)=(y+dy)(z+dz)

=yz+ydz+zdy+dz*dy
=yz+ydz+zdy as last term is v v v small .

Now h'(x) is defined as
((h+dx)-h(x))/dx

What do you see?

4. Mar 16, 2012

### Dramacon

Haha, that was simpler than I thought! :D Thank you so much!

5. Mar 16, 2012

### SteveL27

Oh there's a cool visualization.

Take a rectangle with width w and height h. Draw a picture of it.

Now increase the width by delta-w and the height by delta-h. Draw the picture and you'll see you now have now increased the area wh by the sum of three new rectangles. As delta-w and delta-h get close to zero, the corner rectangle with area (delta-h)(delta-w) gets MUCH smaller than the other two rectangles so you can ignore it ... and voila, you have the product rule.

I googled around and found a picture ...

6. Mar 16, 2012