http://www.ck12.org Chapter 6. Transcendental Functions
Here we use the Chain Rule:
d
dx[ln(cos 5x) 3 ]= 1
cos^35 x·[ 3 (cos 5x) (^2) ·(−sin 5x)·( 5 )]
=cos^135 x·[−15 cos^25 x·sin 5x]
=−15 sin 5cos 5xx
=−15 tan 5x.
Example 4:
Find the derivative ofy=x^3 log 52 x.
Solution:
Here we use the Product Rule along withdxd[logbu] =uln^1 bdudx:
d
dx[x
(^3) log 52 x] =x (^3) ·d
dx[log^52 x]+
d
dx[x
(^3) ]·log 52 x
=x^3 ·xln 5^1 + 3 x^2 ·log 52 x
= x
2
ln 5+^3 x
(^2) log 52 x.
Example 5:
Find the derivative ofy=lnx+x 1 ·
Solution:
We use the Quotient Rule and the natural logarithm rule:
y′=^1 x
x+ 1
·(x+^1 ()(x^1 +) 1 −) 2 (^1 )(x)
=x+x^1 ·(x+^11 ) 2
=x(x^1 + 1 ).
Integrals Involving Natural Logarithmic Function
In the last section, we have learned that the derivative ofy=lnu(x)isy′=u(^1 x).u′(x). The antiderivative is
∫ u′(x)
u(x)dx=ln|u(x)|+C.
If the argument of the natural logarithm isx,thendxd[lnx] = 1 /x,thus