If f'(x) = ex, then find (f (In (In2x))).
Answers
Answer:
Derivation of the Derivative
Our next task is to determine what is the derivative of the natural logarithm. We begin with the inverse definition. If
y = ln x
then
ey = x
Now implicitly take the derivative of both sides with respect to x remembering to multiply by dy/dx on the left hand side since it is given in terms of y not x.
ey dy/dx = 1
From the inverse definition, we can substitute x in for ey to get
x dy/dx = 1
Finally, divide by x to get
dy/dx = 1/x
We have proven the following theorem
Theorem (The Derivative of the Natural Logarithm Function)
If f(x) = ln x, then
f '(x) = 1/x
Examples
Find the derivative of
f(x) = ln(3x - 4)
Solution
We use the chain rule. We have
(3x - 4)' = 3
and
(ln u)' = 1/u
Putting this together gives
f '(x) = (3)(1/u)
3
=
3x - 4
Example
find the derivative of
f(x) = ln[(1 + x)(1 + x2)2(1 + x3)3 ]
Solution
The last thing that we want to do is to use the product rule and chain rule multiple times. Instead, we first simplify with properties of the natural logarithm. We have
ln[(1 + x)(1 + x2)2(1 + x3)3 ] = ln(1 + x) + ln(1 + x2)2 + ln(1 + x3)3
= ln(1 + x) + 2 ln(1 + x2) + 3 ln(1 + x3)
Now the derivative is not so daunting. We have use the chain rule to get
1 4x 9x2
f '(x) = + +
1 + x 1 + x2 1 + x3
Exponentials and With Other Bases
Definition
Let a > 0 then
a x = ex ln a
Examples
Find the derivative of
f (x) = 2x
Solution
We write
2x = ex ln 2
Now use the chain rule
f '(x) = (ex ln 2)(ln 2) = 2x ln 2
Logs With Other Bases
We define logarithms with other bases by the change of base formula.
Definition
ln x
loga x =
ln a
Remark: The nice part of this formula is that the denominator is a constant. We do not have to use the quotient rule to find a derivative
Examples
Find the derivative of the following functions
f(x) = log4 x
f(x) = log (3x + 4)
f(x) = x log(2x)
Solution
We use the formula
ln x
f(x) =
ln 4
so that
1
f '(x) =
x ln 4
We again use the formula
ln(3x + 4)
f(x) =
ln 10
now use the chain rule to get
3
f '(x) =
(3x + 4) ln 10
Use the product rule to get
f '(x) = log(2x) + x(log(2x))'
Now use the formula to get
ln(2x)
log (2x) =
ln 10
The chain rule gives
2 1
f '(x) = log(2x) + x = log(2x) +
(2x) ln 10 ln 10