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Cogito

Multivariable Calculus · Chapter 4 · Lesson 1

Practice Worksheet

Partial Derivatives

Hold one variable still and differentiate the other.

12 problems · about 23 minutes · F-IF.B.6

NameDate

Figure — use these to answer the problems

This is the slice of a surface at a fixed y. The tangent slope shown here is exactly the partial derivative with respect to x.
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y = 1x² + 0x + 2
  • Point(1, 3)
  • Slope of the tangent2
a1
b0
c, set by the fixed value of y2
x, the point of tangency1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f = 3x²y. What is fₓ at (1, 5)?

    Answer

  2. When taking fₓ, what happens to y?

    • It is treated as a constant.
    • It is set to zero.

  3. f = 5xy. What is fₓ at (2, 3)?

    Answer

  4. f = 5xy. What is fᵧ at (2, 3)?

    Answer

  5. f = x² + y³. What is fₓ at (4, 1)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f = x² + y³. What is fᵧ at (4, 1)?

    Answer

  2. f = 7x + 9. What is fᵧ?

    Answer

  3. f = x²y³. What is fₓᵧ at (1, 1)?

    Answer

Stretch Yourself

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

4 problems

  1. Match each notation to what it means.

    Draw a line from each item on the left to its match on the right.

    • fₓ
    • fₓₓ
    • fₓᵧ
    • Differentiate once by x, holding y fixed
    • Differentiate twice by x
    • Differentiate by x, then by y

  2. fₓᵧ = 7 for a smooth function. What is fᵧₓ?

    Answer

  3. The Rate

    f(x, y) = x²y. What is fₓ at the point (3, 2)?

    Answer

  4. The Other Direction

    f(x, y) = x²y. What is fᵧ at the point (3, 2)?

    Answer