What distinguishes the geometric visualization of the sum rule from the product rule?

Conditions

• Comparing linear combination vs multiplicative interaction
• Using geometric models for differentiation

Reasoning, step by step

1. For the sum rule, imagine plotting $y=f(x)$ and $y=g(x)$ separately.
2. Stack the outputs vertically to represent $y=(f+g)(x)$.
3. Observe that a horizontal shift $dx$ creates vertical shifts $df$ and $dg$ that add directly.
4. For the product rule, imagine a rectangle with width $f(x)$ and height $g(x)$.
5. Change inputs slightly to expand the rectangle.
6. Identify the new area added as two long thin strips plus a tiny corner block.
7. Contrast the simple addition of heights (sum) with the complex area decomposition (product).

Example

Script at 109s says: 'For a sum, stack the two output heights... A common input increment produces an exact sum of output increments.' Script at 251s says: 'For a product, imagine a rectangle... The exact area increment is $g\Delta f+f\Delta g+\Delta f\Delta g$: two strips and a small corner.'

Common misconceptions

• Thinking the product rule involves adding derivatives like the sum rule.
• Ignoring the 'corner' term in the product rule geometry until taking the limit.

Watch the explanation

Explore next

• Why does the corner term disappear in the final derivative formula?
• How does constant multiplication fit into these geometric models?