AP Calculus BC 3.6 Calculating Higher- Order Derivatives - MCQs - Exam Style Questions
No-Calc Question
If \(y=f(x)\) is a function such that \(\dfrac{dy}{dx}=\dfrac{2x-y}{x}\), then \(\dfrac{d^{2}y}{dx^{2}}=\ ?\)
(A) \(\dfrac{2y-2x}{x^{2}}\)
(B) \(\dfrac{y-xy}{x^{2}}\)
(C) \(\dfrac{y-x}{x^{2}}\)
(D) \(\dfrac{y}{x^{2}}\)
(B) \(\dfrac{y-xy}{x^{2}}\)
(C) \(\dfrac{y-x}{x^{2}}\)
(D) \(\dfrac{y}{x^{2}}\)
▶️ Answer/Explanation
Correct answer: (A)
\(y’=2-\dfrac{y}{x}\Rightarrow y”=-\left(\dfrac{y}{x}\right)’\!=-\dfrac{x\,y’-y}{x^{2}}=\dfrac{y-x\,y’}{x^{2}}.\)
Substitute \(y’=2-\dfrac{y}{x}\): \(y-x(2-\dfrac{y}{x})=2y-2x\). So \(y”=\dfrac{2y-2x}{x^{2}}.\)