# Logarithmic Differentiation

### Definition - Logarithmic Differentiation

Recalling our derivative of a Natural Logarithms,

( ln(f(x) ) =
( f(x) )
f(x)

Now, if we multiply both sides by f(x) we get

( ln(f(x) )f(x) =
( f(x) )
f(x)
. f(x)

( ln(f(x) )f(x) =
( f(x) )

( f(x) ) =
( ln( f(x) ))f(x)

Definition:

( f(x) ) = f(x) ( ln( f(x) ))

### Beginner Logarithmic Differentiation

Use the Logarithmic Differentiation Formula:

f'(x) = f(x)(ln(f(x))

to find the derivatives of the following functions.

a)f(x) = x2cos(x)

So we use:

f'(x) = f(x)(ln(f(x))

f'(x) = (x2cos(x))(ln(x2cos(x))

Note, one of our rules for logarithms that is import here is:

ln(g(x)h(x)) = ln(g(x) + ln(h(x))

So,

f'(x) = x2cos(x)(ln(x2) + ln(cos(x))

f'(x) = x2cos(x)[(ln(x2)) + (ln(cos(x)))]

Also note, ln(g(x)n) = n.ln(g(x))

f'(x) = x2cos(x)[(2ln(x)) + (ln(cos(x)))]

= x2cos(x) 2.
1
x
+
(-sin(x))
cos (x)
= x2cos(x)
2
x
-
sin(x)
cos(x)
=
x2cos(x)(2)
x
-
x2cos(x)sin(x)
cos(x)

= 2xcos(x) - x2sin(x)

b) f(x) = (x + 1)2(x + 2)3

Using:

f'(x) = f(x)(ln(f(x))

= (x + 1)2(x + 2)3(ln((x + 1)2(x + 2)3)

Remember, ln(g(x)h(x)) = ln(g(x)) + ln(h(x))

And, ln(g(x)n) = n.ln(g(x))

f'(x) = (x + 1)2(x + 2)32ln(x + 1) + 3ln(x + 2)

= (x + 1)2(x + 2)3(2ln(x + 1)) + (3ln(x + 2))

= (x + 1)2(x + 2)3 2.
1
x + 1
+  3.
1
x + 2
=
(x + 1)2(x + 2)3(2)
(x + 1)
+
(x + 1)2(x + 2)3(3)
(x + 2)

= 2(x + 1)(x + 2)3 + 3(x + 1)2(x + 2)2