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Implicit Differentiation

AP Calculus AB· difficulty 3/5

For sin⁡(xy)=x\sin(xy) = x, find dy/dxdy/dx.

  • A

    1+ycos⁡(xy)xcos⁡(xy)\dfrac{1 + y\cos(xy)}{x\cos(xy)}

  • B

    1−ycos⁡(xy)xcos⁡(xy)\dfrac{1 - y\cos(xy)}{x\cos(xy)}

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  • C

    cos⁡(xy)xy\dfrac{\cos(xy)}{xy}

  • D

    cos⁡(xy)\cos(xy)

Explanation

Differentiate: cos⁡(xy)(y+xy′)=1⇒xcos⁡(xy)y′=1−ycos⁡(xy)\cos(xy)(y + x y') = 1 \Rightarrow x \cos(xy) y' = 1 - y \cos(xy).

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