The Product Rule — an Intuition
More visual calculus for you
How do we take the derivative of a product of functions? You may have memorized this formula:
Here’s a visual representation of the above formula. Suppose u and v are functions of time, t. Each function is the side of the black rectangle (bottom left, below). The area of that rectangle is u·v.
Next, we enlarge the rectangle by du in one direction and by dv in the other direction. Each new rectangle has its area determined by two of u, v, du and dv. Summing the three new rectangles gives us an expression for the infinitesimal growth in area, d(u·v):
The infinitesimal time for this evolution to occur is dt. To find the rate of growth at this instant, divide the growth by dt:
The two infinitesimals in the numerator of the final term allows us to eliminate that term. That leaves us with the final expression. Reverting to prime notation:
This is what we had set out to show.
Stay safe!
— Adam
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