Before the final, flavour C

What does a partial derivative measure?

Hold one input of a function of two variables fixed and what is left is a function of one: a partial derivative is the slope of that slice.

Builds on the chain rule and logarithms.

Inputs, L, K: 4.0, 9.0
Output, Q = 4√(LK): 24.00

Step 1 of 5: A map of output

A workshop makes $Q = 4\sqrt{LK}$ units from $L$ hours of labour and $K$ machine hours. Each curve on the map joins the inputs that make one output: $8, 16, 24, 32$ or $40$ . The point uses $L = 4$ and $K = 9$ , on the curve for $24$ .

Predict first. Keep $K = 9$ and double the labour to $L = 8$ . What happens to output?

1. A map of output
A workshop makes $Q = 4\sqrt{LK}$ units from $L$ hours of labour and $K$ machine hours. Each curve on the map joins the inputs that make one output: $8, 16, 24, 32$ or $40$ . The point uses $L = 4$ and $K = 9$ , on the curve for $24$ .
Question: Keep $K = 9$ and double the labour to $L = 8$ . What happens to output?
About $34$ : $4\sqrt{8 \cdot 9} \approx 33.94$ . The point moves right along the line $K = 9$ , past the curve for $32$ but short of $40$ . Holding $K$ at $9$ leaves a function of $L$ alone, $Q = 12\sqrt{L}$ , drawn below.
The figure: Level curves of Q = 4√(LK) for Q = 8, 16, 24, 32 and 40, with labour L across and machine hours K up. A point at L = 8.0, K = 9.0, where Q = 33.94, and a dashed line through it with K held at 9.0. The slice of Q with K held at 9.0: Q = 12.00√L, against L. The point at L = 8.0, Q = 33.94.
2. Slice and slope
The curve below is the slice of $Q$ along $K = 9$ . Its slope at a point is the partial derivative $\partial Q/\partial L$ , the marginal product of labour: extra output per extra hour of labour, with machine hours held fixed. At $L = 4$ it is $3$ .
Drag the point along the line until one more hour of labour adds about $2$ units: $\partial Q/\partial L = 2$ .
At $L = 9$ the slope is $2\sqrt{9/9} = 2$ , down from $3$ at $L = 4$ : each extra hour adds less than the last. On the map, the level curves cross the line further apart as $L$ grows, which is the same fact.
$\frac{\partial Q}{\partial L} = 4\sqrt{K}\cdot\frac{1}{2\sqrt{L}} = 2\sqrt{\frac{K}{L}}$
The figure: Level curves of Q = 4√(LK) for Q = 8, 16, 24, 32 and 40, with labour L across and machine hours K up. A point at L = 9.0, K = 9.0, where Q = 36.00, and a dashed line through it with K held at 9.0. The slice of Q with K held at 9.0: Q = 12.00√L, against L. The point at L = 9.0, Q = 36.00. Its tangent has slope 2.00.
3. The other slice
Now hold the labour at $L = 4$ and change the machine hours: the slice runs up the map, and below it is $Q = 8\sqrt{K}$ . Its slope is $\partial Q/\partial K$ , the marginal product of machine hours.
Question: At $L = 4$ , $K = 9$ , which adds more output: one more hour of labour, or one more machine hour?
An hour of labour. $\partial Q/\partial K = 2\sqrt{4/9} = \tfrac43 \approx 1.33$ , against $\partial Q/\partial L = 3$ . Each partial derivative answers one question: what one input adds while the other stays where it is.
$\frac{\partial Q}{\partial K} = 2\sqrt{\frac{L}{K}}$
The figure: Level curves of Q = 4√(LK) for Q = 8, 16, 24, 32 and 40, with labour L across and machine hours K up. A point at L = 4.0, K = 9.0, where Q = 24.00, and a dashed line through it with L held at 4.0. The slice of Q with L held at 4.0: Q = 8.00√K, against K. The point at K = 9.0, Q = 24.00. Its tangent has slope 1.33.
4. Held means held
A common slip lets the held input move as well, and adds the two rates: $3 + 1.33 = 4.33$ for $\partial Q/\partial L$ . That number is a slope too, but not along the line $K = 9$ .
Question: Along which path through the point does $Q$ change at $4.33$ per hour?
Along the diagonal, in amber: below, $Q$ as both inputs rise together, with slope $4.33$ at the point. The green slice holds $K = 9$ and has slope $3$ . Along the level curve $Q$ does not change at all. In $\partial Q/\partial L$ , $K$ is a constant, so no $\partial Q/\partial K$ part belongs.
The figure: Level curves of Q = 4√(LK) for Q = 8, 16, 24, 32 and 40, with labour L across and machine hours K up. A point at L = 4.0, K = 9.0, where Q = 24.00, and a dashed line through it with K held at 9.0. An amber line runs through the point at 45 degrees, where both inputs rise together. The slice of Q with K held at 9.0: Q = 12.00√L, against L. The point at L = 4.0, Q = 24.00. Its tangent has slope 3.00. An amber curve is Q along the diagonal, with slope 4.33 at the point.
5. Where it is held matters
Held fixed is not the same as left out. $\partial Q/\partial L = 2\sqrt{K/L}$ has $K$ in it: hold the machine hours at another level and the whole slice below changes.
Drag the point up or down until, at $L = 4$ , one more hour of labour adds $2$ units.
At $K = 4$ , $\partial Q/\partial L = 2\sqrt{4/4} = 2$ . With fewer machine hours each hour of labour adds less, and the slice, now $Q = 8\sqrt{L}$ , is flatter everywhere. A partial derivative depends on both inputs, though only one moves along its slice.
The figure: Level curves of Q = 4√(LK) for Q = 8, 16, 24, 32 and 40, with labour L across and machine hours K up. A point at L = 4.0, K = 4.0, where Q = 16.00, and a dashed line through it with K held at 4.0. The slice of Q with K held at 4.0: Q = 8.00√L, against L. The point at L = 4.0, Q = 16.00. Its tangent has slope 2.00.

Step 1 of 5: A map of output

The lesson as text

1. A map of output

2. Slice and slope

3. The other slice

4. Held means held

5. Where it is held matters