Partial Derivatives

Hold one variable still and differentiate the other.

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

What this lesson teaches

The student computes first and second partial derivatives of functions of two variables.

A partial derivative differentiates in one variable while the other is held fixed.
Geometrically it is the slope of the slice of the surface in that direction.
Mixed partials taken in either order agree for the functions you will meet.

Straightforward practice. Get the method working first.

5 problems

f = 3x²y. What is fₓ at (1, 5)?
Answer 30
Why 6 × 1 × 5 = 30.
When taking fₓ, what happens to y?
Answer It is treated as a constant.
Why Held constant, not zeroed.
f = 5xy. What is fₓ at (2, 3)?
Answer 15
Why fₓ = 5y.
f = 5xy. What is fᵧ at (2, 3)?
Answer 10
Why fᵧ = 5x.
f = x² + y³. What is fₓ at (4, 1)?
Answer 8
Why The y³ term vanishes.

The same ideas with more to keep track of.

3 problems

f = x² + y³. What is fᵧ at (4, 1)?
Answer 3
Why 3y² at y = 1.
f = 7x + 9. What is fᵧ?
Answer 0
Why No y appears anywhere.
f = x²y³. What is fₓᵧ at (1, 1)?
Answer 6
Why fₓ = 2xy³, then differentiate by y to get 6xy².

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

4 problems

Match each notation to what it means.
Answer fₓ → Differentiate once by x, holding y fixed; fₓₓ → Differentiate twice by x; fₓᵧ → Differentiate by x, then by y
Why Read the subscripts left to right.
fₓᵧ = 7 for a smooth function. What is fᵧₓ?
Answer 7
Why Clairaut's theorem.
The Rate: f(x, y) = x²y. What is fₓ at the point (3, 2)?
Answer 12
Why 2 × 3 × 2 = 12.
The Other Direction: f(x, y) = x²y. What is fᵧ at the point (3, 2)?
Answer 9
Why 9.