Why does the product rule have two terms?

1. 1. A product is an area

   Draw the product of $u(x) = x$ and $v(x) = x^2$ as a rectangle $u$ wide and $v$ tall: its area is $uv = x^3$. Drag the corner to change $x$, and both sides change with it.

   Question: Make $x$ bigger by $10\%$. By roughly how much does the area grow?

   About $33\%$: $1.1^3 = 1.331$. The width grows by $10\%$ and the height, $x^2$, by $21\%$, and the area takes both. A product changes because each factor does, which is why its derivative cannot be one term.

   The figure: A rectangle u = x = 1.20 wide and v = x² = 1.44 tall, with area 1.728.

2. 2. Nudge it

   Now nudge $x$ by $\Delta x$. The width grows by $\Delta u$ and the height by $\Delta v$, and the new area is the old one plus three pieces: a strip on the right, a strip on top and a small corner. Drag the hollow corner.

   $$\Delta(uv) = v\,\Delta u + u\,\Delta v + \Delta u\,\Delta v$$$\Delta (uv)$ $=v\Delta u+u\Delta v+\Delta u\Delta v$

   The figure: A rectangle 1.20 wide and 1.44 tall, nudged to 1.50 by 2.25: a strip on the right of area 0.432, a strip on top of area 0.972 and a corner of area 0.2430.

3. 3. Shrink the nudge

   A derivative divides the growth by $\Delta x$ and lets $\Delta x$ go to $0$. The numbers now show each piece divided by $\Delta x$.

   Question: As $\Delta x$ shrinks to $0$, which piece, divided by $\Delta x$, goes to $0$?

   The corner. Each strip is about $\Delta x$ thick, so divided by $\Delta x$ it leaves $v\,u' = 1.44$ and $u\,v' = 2.88$. The corner is $\Delta u \cdot \Delta v$, two small numbers multiplied, so divided by $\Delta x$ it is still small and goes to $0$.

   $$(uv)' = u'v + uv'$$$(uv{)}^{\prime }$ $=u^{\prime }v+u{v}^{\prime }$

   The figure: A rectangle 1.20 wide and 1.44 tall, nudged to 1.21 by 1.46: a strip on the right of area 0.014, a strip on top of area 0.029 and a corner of area 0.0002.

4. 4. The slip is the corner

   A common slip differentiates factor by factor: $(uv)' = u'v'$. In the figure, $u'v'$ times $(\Delta x)^2$ is roughly the corner, the one piece that vanishes, while both strips are dropped.

   Question: For $x^3 = x \cdot x^2$ at $x = 1.2$, the slip gives $u'v' = 1 \cdot 2x = 2.4$. What is the true derivative, $3x^2$?

   $3(1.2)^2 = 4.32$, which is the two strips: $v\,u' = 1.44$ plus $u\,v' = 2.88$. The slip's $2.4$ is neither strip; it measures the corner, the piece the limit throws away.

   The figure: A rectangle 1.20 wide and 1.44 tall, nudged to 1.50 by 2.25: a strip on the right of area 0.432, a strip on top of area 0.972 and a corner of area 0.2430.

5. 5. A square grows by two strips

   Make both sides $x$: the rectangle becomes a square of area $x^2$. Nudged, it gains two equal strips, each $x$ by $\Delta x$, and a corner $(\Delta x)^2$.

   Question: So what is the derivative of $x^2$?

   $2x$: two strips of $x$ each, and the corner goes. Drag $x$ and the two strips, divided by $\Delta x$, are always $x$ each. The power rule for $x^2$ is the product rule with $u = v = x$.

   $$\frac{d}{dx}\,x^2 = x + x = 2x$$$\frac{d}{dx}{x}^2$ $=x+x$ $=2x$

   The figure: A rectangle 1.20 wide and 1.20 tall, nudged to 1.21 by 1.21: a strip on the right of area 0.012, a strip on top of area 0.012 and a corner of area 0.0001.

The film, as said, 1:12

1. 0:00

   Draw $u = x$ as a width and $v = x^2$ as a height. Their product is the rectangle's area.

2. 0:05

   Make x bigger and both sides grow, so the area grows on two fronts.

3. 0:13

   Back to $x = 1.2$.

4. 0:16

   Nudge $x$ by $\Delta x$. The new area is the old one plus three pieces.

5. 0:23

   A strip on the right, $v\,\Delta u$. A strip on top, $u\,\Delta v$. And a small corner, $\Delta u\,\Delta v$.

6. 0:28

   A derivative divides the growth by $\Delta x$ and shrinks $\Delta x$ to nothing. The numbers now show each piece divided by $\Delta x$.

7. 0:35

   The strips leave $v\,u'$ and $u\,v'$. The corner, two small numbers multiplied, leaves nothing.

8. 0:44

   So $(uv)' = u'v + uv'$: one term for each side that grows.

9. 0:48

   Differentiating factor by factor gives $u'v'$. In the picture, that is the corner: the one piece that vanishes.

10. 0:55

    Make both sides $x$, and the rectangle becomes a square of area $x^2$.

11. 1:02

    It grows by two strips of $x$ each, and the corner vanishes. So $x^2$ has derivative $2x$.