15.1.1 Definitions and Formulas

Composition of Functions

This is one way of making a new function from two old functions. Essentially, we take one function and ”plug it into” the other function. For example, if we compose f(x) = 2x³ and g(x) = 4x-5 we get either h(x) = (f ∘g)(x) = f(g(x)) = 2(4x - 5)³ or we get k(x) = (g ∘ f)(x) = 4(2x³) - 5 depending on the order of the composition. In general, the two orders are not the same.

Chain rule

We’ll be using this rule a lot. The symbolic analysis section will explain it in more detail, but the basic idea is that if you have a function composed with another function and you need the derivative of the combined object, you use the chain rule to ”chain together” derivatives of each function. For example, if we start with the functions f(x) and g(x) above and compose them into h(x) the new function h is no longer a simple power function or polynomial (although we could multiply it out into a polynomial.) But since it is composed of these simpler functions, we can still take it’s derivative. In fact, the chain rule says that

Thus h′(x) = [df∕dg][dg∕dx] = [2 ⋅ 3g(x)²] ⋅ [4] = 24(4x - 5)². A derivation and proof of the chain rule are somewhat technical; for now, think of this as a way of chaining together the derivatives so the objects which look like (but aren’t really) fractions will cancel out. In the above illustration of the chain rule, the first ”fraction” has the numerator we want (df) and the second ”fraction” has the denominator we want (dx). Each of these ”fractions” has a dg term that ”cancels out” to give the derivative we want: df∕dx.

Product rule

The product rule allows us to take derivatives of functions that are products of simpler functions. It says that

The proof of this rule will be given in the symbolic analysis section, and will make use of the derivative of a logarithm and the chain rule.

Quotient rule

The product rule allows us to take derivatives of functions that are products of simpler functions. It says that

The proof of this rule will be given in the symbolic analysis section, and will make use of the derivative of a logarithm and the chain rule.