Recalling our derivative of a Natural Logarithms,
Now, if we multiply both sides by f(x) we get
Definition:
( f(x) ) = f(x)
( ln( f(x) ))
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)))]
= 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)3 = (x + 1)2(x + 2)3 = 2(x + 1)(x + 2)3 + 3(x + 1)2(x + 2)2
2ln(x + 1) + 3ln(x + 2)
(2ln(x + 1)) +
(3ln(x + 2))
2.