MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A differentiable function f(x)=(g(x)h(x))k(x) is defined for all x with g(x),h(x),k(x)>0. Which of the following statements best explains why setting G(x)=lnf(x) can make it easier to find f′(x)?
(a) Because taking ln turns the complicated form into a sum of three separate functions, whose derivative is simply the sum g′(x)+h′(x)+k′(x).
(b) The natural logarithm transforms products into sums and variable powers into products, so differentiating G requires only sums of simpler terms.
(c) Because ln "cancels" the exponent so that G(x)=k(x), which means f′(x)=k′(x).
(d) Since dxdlnx=1, differentiating G produces the original inside unchanged, giving f′(x)=g(x)h(x).