Practice question · Multiple choice
Differentiating y² with respect to x gives 2y·(dy/dx), not 2y. Why does the extra factor appear?
Hints
- Ask what y depends on. Is it a constant, or is it varying with x?
- Write y² as (y(x))² and apply the chain rule.
Show the answer
A. Because y is a function of x, so y² is a composition.
Why
Implicit differentiation exists for the case where y is defined by an equation you cannot solve, so y is a function of x and y² is really (y(x))², a composition, and dropping the factor treats y as constant, contradicting the setup. Check it on the circle: both routes give −2x, and only because the chain-rule factor was kept.
Practise Related Rates and Implicit Differentiation
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