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.

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

\frac{dz}{dx}& = &\frac{dz}{dy} × \frac{dy}{dx}

= (*e*^{y}) × (-sin x)

= *e*^{{cos x}}(-sin x).

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

(e^{u})' = e^{u} × \frac{du}{dx}

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

(u^{n})' = nu^{{n-1}} × u'

(sin u)' = cos u × u'

(cos u)' = -sin u × u'

(*ln* u)' = 1/u × u' = u'/u

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.

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