1. Derivative of Natural Logs And Logarithmic Differentiation
Lesson 3-9
Derivative of Natural Logs
And
Logarithmic Differentiation

2. Objectives
Know derivatives of regular and natural logarithmic functions
Take derivatives using logarithmic differentiation

3. Logarithmic Functions
Logarithmic Functions: loga x = y  ay = x Cancellation Equations: loga (ax) = x x is a real number a loga x = x x > 0 Laws of Logarithms: loga (xy) = loga x + loga y loga (x/y) = loga x - loga y loga xr = r loga x (where r is a real number)

4. Natural Logs
Natural Logarithms: loge x = ln x ln e = 1 ln x = y  ey = x Cancellation Equations: ln (ex) = x ln e = x x is a real number eln x = x x > 0 Change of Base Formula: loga x = (ln x) / (ln a)

5. Laws of Logs Practice
Simplify the following equations using laws of logarithms
y = ln (12a4 / 5b3)
y = (ln12 + 4lna) – (ln5 + 3lnb)
= ln12 + 4lna – ln5 – 3lnb
y = ln(2a4b7c3)
y = ln2 + 4lna + 7lnb + 3lnc

6. Laws of Logs Practice
Simplify the following equations using laws of logarithms
y = ln[(x²)5(3x³)4 / ((x + 1)³(x - 1)²)]
f(x) = ln[(tan3 2x)(cos4 2x) / (e5x)]

7. Laws of Logs Practice Y = ln a – ln b + ln c Y = 7ln a + 3ln b
Combine into a single expression using laws of logarithms
Y = ln a – ln b + ln c
Y = ln (ac/b)
Y = 7ln a + 3ln b
Y = ln a^2b^3
Y = 3ln a – 5ln c
Y = ln (a^3/b^5)

8. Derivatives of Logarithmic Functions
--- (loga x) =
dx x ln a
d d
--- (loge x) = ---(ln x) = =
dx dx x ln e x
d du u'
--- (ln u) = ----•---- = Chain Rule
dx u dx u d
--- (ln |x|) = (from example 6 in the book)
dx x

9. Example 1 Find second derivatives of the following: 1. f(x) = ln(2x)
2. f(x) = ln(√x)
u = 2x du/dx = 2
d(ln u)/dx = u’ / u
f’(x) = 2/2x
f’(x) = 1/x
u = x du/dx = ½ x-½
d(ln u)/dx = u’ / u
f’(x) = ½ (x-½ ) / x
= 1 / (2 xx)
= 1/2x
f(x) = ½ (ln x)
f’(x) = 1/(2x)

10. Example 2 u = (x² – x – 2) u’ = (2x – 1) 3. f(x) = ln(x² – x – 2)
4. f(x) = ln(cos x)
f’(x) = (2x – 1) / (x² – x – 2)
u = (cos x)
u’ = (-sin x)
f’(x) = (-sin x) / (cos x)
f’(x) = - tan x

11. Example 3 Find the derivatives of the following: 5. f(x) = x²ln(x)
6. f(x) = log2(x² + 1)
Product Rule!
f’(x) = x²(1/x) + 2x ln (x)
= x + 2x ln (x)
Log base a Rule!
d u’
--- (loga u) =
dx u ln a
f’(x) = (2x) / (x² + 1)(ln 2)

12. Summary & Homework Summary: Derivative of Derivatives
Use all known rules to find higher order derivatives