Patterns

Combining the derivatives of basic functions with the chain rule gives us a lot of patterns that let us take derivatives of functions that seem complicated.

Sample Problem

Let h(x) = e^{cos x}. If we think of this as h(x) = f(g(x)) where f(□) = e^□ and g(x) = cos x, 

the chain rule tells us that

h'(x) = f ' (g(x)) · g ' (x)

 = e^{cos x} · (cos x)'

 = e^{cos x} · (-sin x)

If instead we use Leibniz notation, we have z = e^y where y = cos x. The chain rule says

.

Thankfully, we find the same thing either way: the derivative is the original function e^{cos x}, multiplied by the derivative of the power.

We can state this formally as

(e^u)' = e^u · u',

assuming that the prime notation means "take the derivative with respect to x." Using Leibniz notation, we would say

There's also a less formal way that might make more sense:

if h(x) = e^{□}, then the derivative of h is

h ' (x) = e^{□} × (□)'

Similarly, the chain rule tells us that

and so on and so forth. These are good patterns to know, because then we can find derivatives without having to think much about the chain rule.