Chain Rule Derivative

The chain rule differentiates one function inside another. Name the inside u = g(x); the slope of the whole is the slope of the outside with respect to u, times the slope of the inside with respect to x. Worked example: y = (3x + 1)⁵ at x = 1. The inside is u = 3x + 1 = 4 with du/dx = 3; the outside is u⁵ with dy/du = 5u⁴ = 5 × 256 = 1280; so dy/dx = 1280 × 3 = 3840. Leibniz's notation makes the rule look like fractions cancelling, du over du, and that is a fair memory aid for what it says: rates multiply along a chain, the way metres per revolution times revolutions per second gives metres per second.

The classic mistake is to evaluate the outer slope at x instead of at u. For y = sin(x²) at x = 3, dy/du is cos(u) read at u = 9, so the slope is cos(9) × 6, not cos(3) × 6. The other half of the same mistake is dropping the inner factor altogether: 5(3x + 1)⁴ alone is 1280, a third of the true slope. Every related-rates problem is this rule with time as the inner variable, every sine with a frequency inside it and every exponential with a rate in its exponent picks up the inner slope as a factor in front, and backpropagation, the method that trains neural networks, is the chain rule applied through millions of nested functions.