This week was fairly simple. It was easier than most weeks and the topics themselves were easier than most that we've covered. This week, we started taking the derivative of natural log and of some number, we first learned with e raised to an unknown exponent.
First, I’ll talk about taking the derivative of natural log.
When taking the derivative of natural log, drop the ln and change the variable to a fraction allowing the variable to be the denominator and then multiplying the fraction by the derivative of what is inside of natural log. There may also be a product rule, quotient rule, or chain rule within the natural log.
Ex1: ln x = 1/x
Ex2: ln (8x)^2= (1/(8x^2))(16x)= 16x/8x^2
Ex3: (ln 2x)^2= 2(ln 2x)(2)= 2(1/2x)(2)= 4(1/2x)=2/x
Next, I will talk about taking the derivative of e to an unknown exponent n. I would just like to justify that e is in fact a number. It has a numerical value; therefore, you are taking the derivative of a number raised to an unknown exponent n. First, you copy e to n exponent and then you multiply it by the derivative of the exponent. There may also be a product rule, quotient rule, or chain rule within this type of function.
Ex 1: e^x= e^x (1)= e^x
Ex 2: e^(5x)= e^(5x) (5)= 5e^(5x)
Ex 3: e^(x^2)= e^(x^2)(2x)= 2xe^(x^2)
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