11/7/10

Welllllllllllllllll, this week we skipped over to Chapter 5 and learned how to derive natural logs, logs, exponential functions, etc...Thank god we finally had a break from the more difficult things. This week was such a breeze for me; and I understood EVERYTHING! Soooo, with that, here are some examples :)

Ex. 1) Find the derivative of the function:
a.) lnx^2
*the rules for taking the derivative of a natural log say that you have to make it 1/the inside x the derivative
*So you should get 1/x^2(2x)
=2x/x^2..which simplifies to give you 2/x
b.) ln(2x^2+1)
=1/2x^2+1(4x)
=4x/(2x^2+1) ..and that does not simplify so that's your derivative
c.) ln(x+1)^2
*seeing that the function is square should give you the hint that you need to use the chain rule for this...but not yet
*First you should get 1/(x+1)^2
*Now you have to multiply that^ by the derivative of the denominator which is (using the chain rule) 2(x+1)
*So you should get 2(x+1)/(x+1)^2
*the (x+1)'s can cancel and you're left with 2/(x+1)
d.) lnx/x^2
*this is in fact a quotient rule...so here we go
*Using the quotient rule you should end up with this as your first step:
(x^2)(1/x)-[(lnx)(2x) / (x^4) ...keep in mind that the derivative of lnx is 1/x
*simplifying that^ you should get (x-2xlnx)/x^4

Ex. 2) Find an equation of the tangent line to the graph of f at the given point:
f(x)=3x^2-lnx (1,3)
*First take the derivative of the function
*You should get: 6x - (1/x)
*You have the point (1,3)..to find the equation of the tangent line, you have to plug that x-value into your derivative to get your slope
*So you get 6(1) -1/1
=5
*Now you plug everything into the point-slope formula and you should get this:
y-3=5(x-1)

Ex. 3) Find the derivative of the following:
a.) f(x)=e^2x
*In order to find the derivative of anything with e, the rule says that you have to recopy the function and multiply by the derivative (derivative of e's exponent)
*So you should get e^2x(2)
*And that can be written as 2e^2x
b.) 3e^(1-x^2)
*First, like a regular derivative, you would leave the number out in front and then you would recopy the function..then multiply by its derivative
*So you should have:
3(e^1-x^2 x -2x) ....x means times
*And simplifying that you should get -6xe^(1-x^2)
c.) x^2e^-x
*looking at this problem you should notice that two things are being multiplied..sooo, that means you'll have to use the product rule
*So you should get this as your first step: (x^2)(e^-x x -1) + (e^-x)(2x)
*Simplifying that you get -x^2e^-x + 2xe^-x
d.) 7^(2x-1)
*hold upppppp, there's no e in this one so you can't take the derivative the same way you have been for the other ones..there's a separte rule for this!
*it says that you recopy the function, multiply it by "ln#", and also multiply it by the functions derivative. that probably didn't make sense, but here's what I mean..
*You should get 7^(2x-1) x ln7 x 2
*Well you can't really simplify anything I don't think so you can just rewrite it as:
2(ln7)7^(2x-1)

**Like I said before, this week was easyyyyy and I had no problems!