A metal rod is heated so that its length increases while its mass remains constant
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A metal rod is heated so that its length L(t) increases while its mass remains constant. To model the changing density ρ(t), a student writes ρ(t)L(t)=k, where k is a positive constant. The student differentiates both sides with respect to t \,without first solving for ρ.
Which of the following best justifies that this step is mathematically valid?
(a) Implicit differentiation is permissible only after the dependent variable is isolated, so differentiating ρL=k at this stage is invalid.
(b) Differentiating an equation preserves equality even without isolating variables, so implicit differentiation of ρL=k is valid.
(c) Because the product ρL is constant, both ρ and L must themselves be constant, so their derivatives are zero and differentiation is unnecessary.
(d) Differentiating a product means differentiating each factor separately, so treating dtd(ρL) as a single derivative is incorrect; the student should avoid differentiating until ρ is alone.