Logarithmic Differentiation

Definition - Logarithmic Differentiation

Recalling our derivative of a Natural Logarithms,

d over dx derivative operator( ln(f(x) ) = 
d over dx derivative operator( f(x) )
f(x)

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

opening square bracketd over dx derivative operator( ln(f(x) )closing square bracketf(x) = opening square bracket
d over dx derivative operator( f(x) )
f(x)
closing square bracket . f(x)

opening square bracketd over dx derivative operator( ln(f(x) )closing square bracketf(x) = 
d over dx derivative operator( f(x) )

d over dx derivative operator( f(x) ) = 
opening square bracketd over dx derivative operator( ln( f(x) ))closing square bracketf(x)

Definition:

d over dx derivative operator( f(x) ) = f(x)opening square bracket d over dx derivative operator( ln( f(x) )) closing square bracket

Beginner Logarithmic Differentiation

Use the Logarithmic Differentiation Formula:

f'(x) = f(x)d over dx derivative operator(ln(f(x))

to find the derivatives of the following functions.

a)f(x) = x2cos(x)

So we use:

f'(x) = f(x)d over dx derivative operator(ln(f(x))

f'(x) = (x2cos(x))d over dx derivative operator(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)d over dx derivative operator(ln(x2) + ln(cos(x))

f'(x) = x2cos(x)[d over dx derivative operator(ln(x2)) + d over dx derivative operator(ln(cos(x)))]

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

f'(x) = x2cos(x)[d over dx derivative operator(2ln(x)) + d over dx derivative operator(ln(cos(x)))]

= x2cos(x)opening square bracket opening bracket2.
1
x
closing bracket +  
(-sin(x))
cos (x)
closing square bracket
= x2cos(x) opening bracket
2
x
  -  
sin(x)
cos(x)
closing bracket
=
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)d over dx derivative operator(ln(f(x))

= (x + 1)2(x + 2)3d over dx derivative operator(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)3opening square bracketd over dx derivative operatoropening bracket2ln(x + 1) + 3ln(x + 2)closing bracket closing square bracket

= (x + 1)2(x + 2)3opening square bracketd over dx derivative operator(2ln(x + 1)) + d over dx derivative operator(3ln(x + 2)) closing square bracket

= (x + 1)2(x + 2)3 opening square bracket2.
1
x + 1
 +  3.
1
x + 2
closing square bracket
=
(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

 

Logarithmic Differentiation

First part of worked example showing logarithmic differentiation used to differentiate f(x) = (5x + 3)^4 multiplied by sin(2x)
Second part of worked example showing logarithmic differentiation used to differentiate f(x) = (5x + 3)^4 multiplied by sin(2x)

Logarithmic Differentiation

First part of worked example showing logarithmic differentiation used to differentiate f(x) = (5x - 1)^6 multiplied by (x + 4)^2 multiplied by (2(x^2) + 5)^2
Second part of worked example showing logarithmic differentiation used to differentiate f(x) = (5x - 1)^6 multiplied by (x + 4)^2 multiplied by (2(x^2) + 5)^2

Logarithmic Differentiation

First part of calculation showing how to use logarithmic differentiation to find the derivative of x^x
Second part of calculation showing how to use logarithmic differentiation to find the derivative of x^x

Logarithmic Differentiation

First part of calculation showing how to use logarithmic differentiation to find the derivative of x^(cos(x))
Second part of calculation showing how to use logarithmic differentiation to find the derivative of x^(cos(x))