Implicit Differentiation and Derivatives of Inverse Functions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Slopes without solving for y

Mathematics · Calculus & Analysis · ages 18-19
Name ______________________   Date ____________

  1. True or false: when you differentiate y^2 with respect to x, the result is 2y times dy/dx.

    Circle one:   True   False

  2. What is the derivative of arcsin x with respect to x?

    • 1/sqrt(1 - x^2)
    • -1/sqrt(1 - x^2)
    • 1/(1 + x^2)
    • sqrt(1 - x^2)

  3. Differentiating y squared with respect to x gives 2y times dy/dx.

    Circle one:   True   False

  4. What is the derivative of arccos x?

    • 1/sqrt(1 - x squared)
    • -1/(1 + x squared)
    • -1/sqrt(1 - x squared)

  5. Let y = arcsin x. Find the value of dy/dx at x = 0.

    Answer: ______________

  6. What is the derivative of arctan(3x)?

    • 3/(1 + 9x^2)
    • 1/(1 + 9x^2)
    • 3/(1 + x^2)
    • 9x/(1 + 9x^2)

  7. For xy = 12 at (3, 4), find dy/dx to two places.

    Answer: ______________

  8. True or false: differentiating sin y = x with respect to x gives cos y times dy/dx = 1, so dy/dx = 1/cos y.

    Circle one:   True   False

  9. The curve x^2 + xy + y^2 = 37 passes through the point (3, 4). Find dy/dx there. Give your answer as a decimal rounded to two places.

    Answer: ______________

  10. For x squared + xy + y squared = 37 at (3, 4), find dy/dx to two places.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Slopes without solving for y W1-mt_P4ABBGsKFT-s1

  1. True · True. y is a function of x, so the chain rule multiplies the outer derivative 2y by the inner derivative dy/dx.
  2. 1/sqrt(1 - x^2) · The derivative of arcsin x is 1/sqrt(1 - x^2). You get it by writing sin y = x and differentiating implicitly.
  3. True · Y depends on x, so the chain rule multiplies the outer 2y by dy/dx.
  4. -1/sqrt(1 - x squared) · It matches arcsin with a minus sign, from differentiating cos y = x.
  5. 1 · Since dy/dx = 1/sqrt(1 - x^2), plugging in x = 0 gives 1/sqrt(1) = 1.
  6. 3/(1 + 9x^2) · The derivative of arctan u is u'/(1 + u^2). With u = 3x this gives 3/(1 + 9x^2).
  7. -1.33 · Product rule gives dy/dx = -y/x = -4/3, about -1.33.
  8. True · True. The chain rule turns the derivative of sin y into cos y times dy/dx, and dividing by cos y solves for dy/dx.
  9. -0.91 · Implicit differentiation gives dy/dx = -(2x + y)/(x + 2y), and at (3, 4) that is -10/11, about -0.91.
  10. -0.91 · Collecting gives -(2x + y)/(x + 2y) = -10/11, about -0.91.
Worksheet · LightMySky