Today I saw this beautiful and intuitive explanation of the product rule on Quora.

Consider a rectangle with sides f(x) and g(x)

The area of the square, A(x) = f(x)g(x)

Now, d(f(x)g(x))/dx = d(A(x))/dx

Let us increase x by an amount dx

The new area is given as A(x + dx) = f(x + dx)g(x + dx)

Now note that f(x + dx) = f(x) + d(f(x))

Also, g(x + dx) = g(x) + d(g(x))

So the new area is A(x + dx) = (f(x) + d(f(x)))(g(x) + d(g(x)))

= A(x) + f(x)d(g(x)) + g(x)d(f(x)) + d(f(x))d(g(x))

Change in area = A(x + dx) - A(x) = f(x)d(g(x)) + g(x)d(f(x)) + d(f(x))d(g(x))

Divide throughout by dx and ignore the last term (because its a product of differentials), you get

(A(x + dx) - A(x)) / dx = f(x)d(g(x))/dx + g(x)d(f(x))/dx

The left hand side by definition is d(A(x))/dx and the right hand side is f(x)g'(x) + g(x)f'(x)

This picture sums it all

Profound!

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