Apentix/AP Calculus AB/Unit 3: Differentiation: Composite, Implicit, and Inverse Functions/Question
AP Calculus ABmediummcq1 pt

If sin⁡(xy)=x\sin(xy)=x defines yy implicitly as a function of xx, then dydx=\dfrac{dy}{dx}=

A.sec⁡(xy)x\dfrac{\sec(xy)}{x}
B.cos⁡(xy)−yx\dfrac{\cos(xy)-y}{x}
C.sec⁡(xy)−yx\dfrac{\sec(xy)-y}{x}
D.sec⁡(xy)+yx\dfrac{\sec(xy)+y}{x}

Explanation

Core Concept

Differentiating gives cos⁡(xy)(y+xdydx)=1\cos(xy)\left(y+x\frac{dy}{dx}\right)=1. Dividing by cos⁡(xy)\cos(xy) and solving gives dydx=sec⁡(xy)−yx\frac{dy}{dx}=\frac{\sec(xy)-y}{x}.

Correct Answer

Csec⁡(xy)−yx\dfrac{\sec(xy)-y}{x}

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