Calculus Volume 1 · 3.8 Implicit Differentiation

Implicit Differentiation: practice problems

17 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 301

   For the following exercises, use implicit differentiation to find $\frac{dy}{dx}.$

   $6x^2+3y^2=12$

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2. Exercises 303

   For the following exercises, use implicit differentiation to find $\frac{dy}{dx}.$

   $3x^3+9x{y}^2=5x^3$

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3. Exercises 305

   For the following exercises, use implicit differentiation to find $\frac{dy}{dx}.$

   $y\sqrt{x+4}=xy+8$

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4. Exercises 307

   For the following exercises, use implicit differentiation to find $\frac{dy}{dx}.$

   $y\sin (xy)=y^2-2$

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5. Exercises 309

   For the following exercises, use implicit differentiation to find $\frac{dy}{dx}.$

   $x^{3}y+x{y}^3=-8$

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6. Exercises 311

   For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

   [T] $x^2{y}^2+5xy=14,(2,1)$

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7. Exercises 313

   For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

   [T] $x{y}^2+\sin (\pi y)-2x^2=10,(2,-3)$

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8. Exercises 315

   For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

   [T] $xy+\sin (x)=1,(\frac{\pi }{2},0)$

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9. Exercises 317

   For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

   For the equation $x^2+2xy-3y^2=0,$

   1. Find an equation of the normal to the tangent line at the point $(1,1).$
   2. At what other point does the normal line in a. intersect the graph of the equation?

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10. Exercises 319

    For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

    For the equation $x^2+xy+y^2=7,$

    1. Find the $x$-intercept(s).
    2. Find the slope of the tangent line(s) at the x-intercept(s).
    3. What does the value(s) in b. indicate about the tangent line(s)?

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11. Exercises 321

    For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

    Find an equation of the tangent line to the graph of the equation ${\tan }^{-1}(x+y)=x^2+\frac{\pi }{4}$ at the point

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12. Exercises 323

    For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

    [T] The number of cell phones produced when $x$ dollars is spent on labor and $y$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $60x^{3\text{/}4}{y}^{1\text{/}4}=3240.$

    1. Find $\frac{dy}{dx}$ and evaluate at the point

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13. Exercises 325

    For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

    The volume of a right circular cone of radius $x$ and height $y$ is given by $V=\frac{1}{3}\pi x^{2}y.$ Suppose that the volume of the cone is a constant. Find $\frac{dy}{dx}$ when and

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14. Exercises 326

    For the following exercises, consider a closed rectangular box with a square base with side $x$ and height $y.$

    Find an equation for the surface area of the rectangular box, $S(x,y).$

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15. Exercises 327

    For the following exercises, consider a closed rectangular box with a square base with side $x$ and height $y.$

    If the surface area of the rectangular box is 78 square feet, find $\frac{dy}{dx}$ when $x=3$ feet and $y=5$ feet.

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16. Exercises 329

    For the following exercises, use implicit differentiation to determine $y^{\prime }.$ Does the answer agree with the formulas we have previously determined?

    $x=\cos y$

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17. Review Exercises 385

    Find an equation of the tangent line to the following equations at the specified point.

    $y=x+e^x-\frac{1}{x}$ at $x=1$

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