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For this blog I'm going to review Chapter 5..taking derivatives of logs, natural logs, e, exponents and whatnot. *first let's start with natural logs. The rule for taking the derivative of those is to first write the inside as a fraction over 1 and then multiply it by its derivative. So it would be like this: 1/# x d/dx of # So here are some examples like that: 1.) ln(5x+4) =1/5x+4 x 5 =5/5x+4 2.) ln(5x^2+2x+4) =1/(5x^2+2x+4) x 10x+2 =(10x+2)/(5x^2+2x+4) 3.) ln(x^2-16)^1/2 =1/(x^2-16)^1/2 x 1/2(x^2-16)^-1/2 x 2x =x/(x^2-16) *Now let's go over how to derive exponential functions dealing with e. The rule says that all you do to take the derivative is first recopy the function they give you, then multiply it by the exponent's derivative So here are some examples: 1.) f(x)=(4e)^-3x^2 -First instead of recopying the whole thing, you leave the 4 out in front just like you would do when deriving any other equation. Then you recopy the e^-3x^2 -So you should have 4(e^-3x^2 -Now multiply that by the derivative of e^-3x^2 which is -6x -So now you have 4(e^-3x^2 x -6x) = -24xe^-3x^2 2.) f(x)=x^4e^x -Okay first thing you should realize is that this is a product rule. The same rules apply even though there's an e involved..and to save time, the derivative of e^x is e^x -So using the product rule you get: (x^4)(e^x)+(e^x)(4x^3) =x^4e^x + 4x^3e^x -Then you can simplify that^ further by taking out an x^3 and an e^x -So your final answer is: x^3e^x(x+4) *Last, let's go over how to derive logs. You take the inside of the function, put it as a fraction over 1 times ln(of whatever base) times the derivative of the inside..I'm not sure if I explained that right haha, but anyway here's an example of that: 1.) log (base 3) (6x^2+5x+2) =1/(6x^2=5x+2)ln3 x 12x+5 = (12x+5)/(6x^2+5x+2)ln3