Blog #17

For this holiday blog, I'm going to review Ch.5. It’s very simple, just derivatives of natural logs (ln).

Derivatives of ln:
d/dx ln u = (1/u)(du/dx)
*don’t forget ln properties, product rule, quotient rule, and chain rule

Ex. 1) d/dx ln 3x = (1/3x)(3) = 1/x

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

Ex. 3) d/dx xlnx = (x)(1/x) + (lnx)(1) = 1 + lnx

Ex. 4) d/dx (ln x)^4 = (4(ln x)^3)(1/x)(1) = 4(ln x)^3 / x

Ex. 5) d/dx ln square root (x + 1) = ln(x + 1)^1/2 = ½ ln(x + 1) = (1/2)(1/(x + 1))(1) = 1 / 2(x + 1)

Ex. 6) d/dx ln (2x/3x^2) = ln (2x) – ln (3x^2) = (1/2x)(2) – (1/(3x^2))(6x) = (1/x) – (2/x) = - 1/x

I'm also going to review how to take the derivative of exponential functions.

Derivates of exponential functions:
d/dx e^u = (e^u)(du/dx)

Ex. 7) e^2x = (e^2x)(2) = 2e^2x

Ex. 8) e^(-5/x) = (e^(-5/x))(5x^-2) = 5e^(-5/x) / x^2

Also, how to solve for x in ln equations.

Ex. 9) ln e^x = 6
x = 6

Ex. 10) e^(lnx^2) = 9
x^2 = 9
x = 3, -3

Ex. 11) e^x = 3
ln e^x = ln 3
x = ln 3

Ex. 12) ln (x – 2) = 3
e^2 = x – 3
x = e^2 + 3