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#1 2006-11-29 15:10:27

fusilli_jerry89
Member
Registered: 2006-06-23
Posts: 86

Derivatives

I really have no idea how to do any of these. If some one could show me by example maybe I could catch on?

Find the derivatives:

y=e^-x
y=xe^2-e^x
y=e^sqrtx
y=8^x
y=x^lnx
y=ln(1/x)
y=log(4)x²             [the first bracket means base]
y=log(2)(3x+1)
y=log(10)e^x
y=(sinx)^x, 0<x<π/2

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#2 2006-11-30 06:04:45

polylog
Member
Registered: 2006-09-28
Posts: 162

Re: Derivatives

Here is the 3rd one:

The key to this is the chain rule, and knowing that e^x has the special property:

Thus:

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#3 2006-11-30 06:21:49

polylog
Member
Registered: 2006-09-28
Posts: 162

Re: Derivatives

Here is the 4th one:

There is a really useful formula that can help with this. Let's derive it:

First, there is a crucial result that for all functions f:

So, we can write:

This is just e raised to a constant times x.

Taking the derivative:

But now this can be re-written as:

In general, it's better to just remember the formula:

And notice this still holds if b = e:

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#4 2006-11-30 06:30:42

polylog
Member
Registered: 2006-09-28
Posts: 162

Re: Derivatives

Here is the 7th one:

The key thing here is the Change of Base Formula:


Thus we should rewrite the function like this:

The last step was just to emphasize that 1/ln4 is just a constant multiplier.

Taking the derivative (Chain Rule) :

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