Practice question · Multiple choice
The derivative of a sum is the sum of the derivatives, but the derivative of a product is not the product of the derivatives. Why does one operation pass straight through differentiation and the other not?
Hints
- Test it: differentiate x·x as a product and compare with (x')·(x').
- If both factors change, how many ways does the product change?
Show the answer
D. Because differentiation is linear and multiplication is not.
Why
The counterexample is immediate: f = g = x gives (x²)' = 2x while the product of the derivatives is 1. Picture a rectangle with sides f and g, growing both adds a strip along the top and one along the side, f'g and fg', plus a corner square that vanishes in the limit. That corner is why the naive guess fails and the answer has two terms.
Practise Differentiation Rules
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