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Computing Derivatives

Computing Derivatives


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 = ey where y = cos x. The chain rule says

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

 = (ey) × (-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

(eu)' = eu × u',

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

(eu)' = eu × \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

(un)' = 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.