Logs and Derivatives
III. Definition of the Natural Logarithm
What is ?
Definition: For x > 0 we define
Note: The 2nd FTC tells us that d/dx(lnx) = 1/x
IV. Properties of lnx
1) ln(1) = 0
2) ln(ab) = ln(a) + ln(b)
3) ln(an) = nlna
4) ln(a/b) = lna - lnb
Proof of 3):
So that ln(xn) and nlnx have the same derivative. Hence, n(xn) = nlnx + C. Plugging in x = 1 we have that C = 0.
V) Definition of e
Let e be such that lne = 1 ie.
VI) Examples:(we will work on the following in groups)
Find the derivatives of the following functions:
A) ln(x2 + 1)
G) Show that 3lnx - 4 is a solution of the differential equation
xy'' + y' = 0
H) Find the relative extrema of xlnx
I) Find the equation of the tangent line to 3x2 - lnx at (1,3)
J) Find dy/dx for ln(xy) + 2x2 = 30.