1. The Setting: Why We Need a New Rule
1.1 Composite Functions Recalled
In the functions block you learned that a composite function is formed when the output of one function becomes the input of another. If $f$ and $g$ are functions, the composite $h = f \circ g$ is defined by
$$ h(x) = f\!\bigl(g(x)\bigr). $$For example, $h(x) = \sin(x^3)$ is the composition of $f(u) = \sin u$ (outer) with $g(x) = x^3$ (inner). Similarly, $k(x) = \sqrt{3x^2+1}$ is $f(u) = \sqrt{u}$ composed with $g(x) = 3x^2+1$.
1.2 Why Existing Rules Fall Short
Try differentiating $h(x) = (3x+1)^5$ using only the power rule. You would need to expand the fifth power first — tedious and error-prone. For $\sin(x^3)$ there is no expansion at all. We need a dedicated rule for compositions.
1.3 Numerical Evidence — Rates Multiply Through a Chain
Before stating the rule formally, let's build intuition with everyday "chain" reasoning:
Gear A turns 3 times for every turn of the input shaft. Gear B turns 2 times for every turn of Gear A. How fast does Gear B turn relative to the input?
$\text{Rate}_{B/\text{input}} = \underbrace{3}_{\text{A per input}} \times \underbrace{2}_{\text{B per A}} = 6$
The rates multiply through the chain of dependencies.
A car travels at 60 miles per hour. There are 1.609 km per mile. The speed in km/h is: