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Cogito

AP Calculus AB · Chapter 3 · Lesson 3

Practice Worksheet

Implicit Differentiation

Differentiating an equation you cannot solve.

12 problems · about 22 minutes · FUN-3.D, FUN-3.E

NameDate

Figure — use these to answer the problems

A circle is the standard implicit curve. No single function of x describes both halves at once.
d = 10
  • Diameter10 cm
  • Circumference31.42 cm
  • Circumference ÷ diameter3.14
Radius5
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. For x² + y² = 25, what is the slope at (4, 3), as a decimal to two places?

    Answer

  2. Why does differentiating y² give 2y · dy/dx?

    • y is a function of x, so the Chain Rule supplies its derivative.
    • It is a separate rule for implicit equations.

  3. Differentiating y² with respect to x gives 2y times what?

    • dy/dx
    • One

  4. For x² + y² = 25, slope at (0, 5)?

    Answer

  5. For x² + y² = 25, slope at (3, −4), as a decimal?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. x² + y² = 169. Is (5, 12) on the curve? 1 yes, 0 no.

    Answer

  2. For x² + y² = 169, slope at (5, 12), as a decimal to two places?

    Answer

  3. What is the derivative of a constant?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the implicit differentiation process in order.

    Write 1 to 4 in the boxes to put these in order.

    • Differentiate both sides with respect to x.
    • Attach dy/dx to every term that came from a y.
    • Collect the dy/dx terms on one side.
    • Factor out dy/dx and divide.

  2. For x² + y² = 100, slope at (6, 8), as a decimal?

    Answer

  3. The Slope

    For x² + y² = 25, dy/dx = −x ÷ y. What is the slope at (3, 4), as a decimal?

    Answer

  4. The Other Point

    Same curve. What is the slope at (4, 3), as a decimal to two places?

    Answer