Implicit Differentiation Practice

Sample questions

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1. Sample 1difficulty 2/5

   For $2x + 3y = 6$, $dy/dx =$

   • A

     $-2/3$

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   • B

     $-3/2$

   • C

     $3/2$

   • D

     $2/3$

   Why

   Differentiate: $2 + 3 y' = 0 \Rightarrow y' = -2/3$.

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2. Sample 2difficulty 3/5

   The tangent to $x^2 + y^2 = 25$ at $(3, 4)$ is

   • A

     $y = (3/4)x + 4$

   • B

     $y = -(3/4)x + 25/4$

   • C

     Both B and C

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   • D

     $3x + 4y = 25$

   Why

   Slope $-3/4$. $y - 4 = -3/4 (x - 3) \Rightarrow y = -(3/4)x + 25/4$, equivalently $3x + 4y = 25$.

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3. Sample 3difficulty 3/5

   For $xy = 6$, $dy/dx =$

   • A

     $-x/y$

   • B

     $-y/x$

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   • C

     $y/x$

   • D

     $x + y$

   Why

   Differentiate: $y + x y' = 0 \Rightarrow y' = -y/x$.

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4. Sample 4difficulty 3/5

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

   • A

     $\dfrac{1 + y\cos(xy)}{x\cos(xy)}$

   • B

     $\dfrac{1 - y\cos(xy)}{x\cos(xy)}$

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   • C

     $\dfrac{\cos(xy)}{xy}$

   • D

     $\cos(xy)$

   Why

   Differentiate: $\cos(xy)(y + x y') = 1 \Rightarrow x \cos(xy) y' = 1 - y \cos(xy)$.

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5. Sample 5difficulty 3/5

   For $x^2 + y^2 = 25$, vertical tangents occur where the denominator of $y' = -x/y$ is zero, i.e. $y = 0$. The points are

   • A

     $(0, 0)$

   • B

     $(\pm 3, 4)$

   • C

     $(0, \pm 5)$

   • D

     $(\pm 5, 0)$

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   Why

   $y = 0$ on the circle gives $x = \pm 5$.

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