1. 3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus 14th Edition
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AP Calculus AB - Worksheet 27 Derivatives of ln and e Know the following theorems: 1. The derivative of the natural logarithmic function (lnx) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus.

Related Pages
Natural Logarithm
Logarithmic Functions
Derivative Rules
Calculus Lessons

Natural Log (ln)

The Natural Log is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828. The natural logarithm is usually written ln(x) or log_e(x).

The natural log is the inverse function of the exponential function. They are related by the following identities:
e^ln(x) = x
ln(e^x) = x

Derivative Of ln(x)

Using the Chain Rule, we get

Example:
Differentiate y = ln(x² +1)

Solution:
Using the Chain Rule, we get

Example:
Differentiate

Solution:

Derivatives Of Logarithmic Functions

The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus.

What Are The Formulas For Finding Derivatives Of Logarithmic Functions And How To Use Them To Find Derivatives?

The following are the formulas for the derivatives of logarithmic functions:

Examples:
Find the derivatives for the following logarithmic functions:

1. f(x) = ln(x² + 10)
2. f(x) = √x ˙ ln(x)
3. f(x) = ln[(2x + 1)³/(3x - 1)⁴]
4. y = [log_a(1 + e^x)]²

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Derivatives Of Logarithmic Functions

Find the derivatives for the following logarithmic functions:

Examples:

1. y = ln(x² x)
2. y = (log₇ x)^1/3
3. y = ln(x⁴˙sin x)
4. y = lnx/[1 + ln(2x)]

Derivatives Of The Natural Log Function (Basic)

How to differentiate the natural logarithmic function?

Examples:
Determine the derivative of the function.

1. f(x) = 2ln(x)
2. f(x) = ln(4x)

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Derivatives Of The Natural Log Function With The Chain Rule

How to differentiate the natural logarithmic function using the chain rule?

Example:
Determine the derivative of the function.
f(x) = 5ln(x³)

3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus 14th Edition

The Derivative Of The Natural Log Function

We give two justifications for the formula for the derivative of the natural log function. If you want to see where this formula comes from, this is the video to watch.

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The Derivative of the Natural Logarithm

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

e^y = 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.

e^y dy/dx = 1

From the inverse definition, we can substitute x in for e^y to get

x dy/dx = 1

Finally, divide by x to get

dy/dx = 1/x

We have proven the following theorem

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

3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus Solver

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

Examples
Find the derivative of

f (x) = 2^x

Solution

We write
2^x = e^{x ln 2}

Now use the chain rule
f '(x) = (e^{x ln 2})(ln 2) = 2^x ln 2

3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus Calculator

Logs With Other Bases

We define logarithms with other bases by the change of base formula.

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

3.4 Derivative Of E^f(x) And Ln (f(x))ap Calculus 2nd Edition

Examples
Find the derivative of the following functions

1. f(x) = log₄ x
   

2. f(x) = log (3x + 4)
   

3. f(x) = x log(2x)
   

Solution

1. We use the formula
   ln x
   f(x) =
   ln 4
   
   so that
   1
   f '(x) =
   x ln 4
   
   

2. 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
   
   

3. 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