12.2 Differentiation in practice

During our first encounter with differentiation, we saw that computing derivatives by the definition

can be really hard in practice if we encounter convoluted functions such as f(x) = cos(x)sin(e^x). Similar to convergent sequences and limits, using the definition of differentiation won’t get us far—the complexity piles on fast. So, we have to find ways to decompose the complexity into its fundamental building blocks.

12.2.1 Rules of differentiation

First, we’ll look at the simplest of operations: scalar multiplication, addition, multiplication, and division.

Theorem 80. (Rules of differentiation)

Let f : ℝ → ℝ and g : ℝ → ℝ be two arbitrary functions and let x ∈ ℝ. Suppose that both f and g is differentiable at x. Then

(a) (cf)^′(x) = cf^′(x) for all c ∈ℝ,

(b) (f + g)^′(x) = f^′(x) + g^′(x),

(c) (fg)^′(x) = f^′(x...