Derivatives

fusilli_jerry89 · 2006-11-29 15:10:27

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

polylog · 2006-11-30 06:04:45

Here is the 3rd one:

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

Thus:

polylog · 2006-11-30 06:21:49

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:

polylog · 2006-11-30 06:30:42

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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