Chain Rule of Differentiation

By the chain rule (f∘g)′(x) = f′(g(x))·g′(x). So you differentiate the outer function and insert the unchanged inner function into it, then multiply by the derivative of the inner function. Remember it as outer derivative times inner derivative. Example: for e^(3x²) the outer function is e to the power of something with derivative e^(3x²), the inner function is 3x² with derivative 6x. The result is therefore 6x·e^(3x²). The most important step is not to forget the inner derivative; it is the most common reason for wrong results with composite functions.

In a composite function f(g(x)), g is the inner function, because it acts on x first, and f is the outer function, because it is applied to the result of g. You recognize the inner function because it sits in the argument of the outer one, that is in the bracket or under the function symbol. In sin(x²), x² is inside and the sine outside. In √(2x + 1), 2x + 1 is inside and the root outside. A helpful question is: what would I enter first into the calculator? This quantity computed first is the inner function. Its derivative provides the decisive factor of the chain rule.

Because a change in x first changes the inner quantity and this change is then passed on through the outer function. The chain rule dy/dx = dy/du · du/dx shows that the two rates of change multiply. If you omit the inner derivative, you only account for how strongly the outer function reacts to its argument, but not how fast the argument itself changes with x. Example: for sin(3x) the derivative is 3·cos(3x), not just cos(3x); the factor 3 comes from the inner derivative. Without it the result would be too small by exactly this factor. That is why the inner derivative is indispensable.

For multiply nested functions you apply the chain rule repeatedly, so to speak from outside to inside. For f(g(h(x))) the derivative is f′(g(h(x)))·g′(h(x))·h′(x). So you differentiate each layer individually and multiply all inner derivatives one after another. Example: sin(√(x² + 1)). The outermost is the sine, then the root, then x² + 1. The derivative is cos(√(x² + 1))·(1/(2√(x² + 1)))·2x. Work systematically from the outermost function inward and multiply all factors. If you forget a layer, a factor is missing and the result is wrong.

For e^(kx) the outer function is the exponential function, whose derivative is again e^(kx), and the inner function is kx with derivative k. By the chain rule you multiply both and get k·e^(kx). The constant factor k from the exponent therefore comes to the front as a prefactor. For e^(3x) the derivative is 3·e^(3x), for e^(−0.5x) you get −0.5·e^(−0.5x). This result is the basis of all growth and decay processes: the derivative is proportional to the function itself, with the rate k as the proportionality factor. That is exactly why the exponential function describes exponential growth and radioactive decay so naturally.