Blog 11/7/2010--Stephen

This week was a pretty good one. It went by pretty fast and what we learned is actually easy and I can speed through the problems lol. This week we learned how to take the derivatives of natural logs and logs, and we also reviewed some of the basic properties of logs and natural logs. I will describe how to take some derivatives of natural logs and logs, and I will also give many examples.

derivative of a natural log:

d/dx ln u = (1/u)(du/dx)

Example:

d/dx ln (2x) = (1/2x)(2) = (1/x)

d/dx ln (x^2 + 1) = ((1)/(x^2+1)) x (2x) = (2x)/(x^2 + 1)

Exponential functions:

number raised to a variable

d/dx e^u = e^u x du/dx

Example:

d/dx e^(2x-1) = e^(2x-1) x 2

2e^(2x-1)

d/dx e^(-3/x) = e^(-3/x) x (3/x^2)

(3e^(-3/x))/(x^2) = ((3)/((x^2)(e^(3/x)))

ln e^x = 4

x = 4

e^ln(x^2) = 9

x^2 = 9

x = plus or minus 3

e^x = 5

ln e^x = ln 5

x = ln 5

ln(x-2) = 3

e^3 = x-2

x = e^3 + 2