The Chain Rule Part 2

We can use the Chain Rule to derive exponential functions, natural logarithms and logarithms. In this lesson we will also start applying it to finding slopes in graphs of equations.

Understanding exactly what logarithms are will help understand how we differentiate them. If its been a while since you have dealt with logarithms, review below,

The natural logarithms of a function, f(x),  is symbolized by, 

ln(f(x)) / or in calligraphy form 𝓵𝓷(f(x)) 

or loge(f(x)) 

and means the inverse to the exponential number, e. The natural logarithm is often just referred to as the logarithm. 

The different representations of logarithms and what they mean can be confusing so we will go over some of the basic ideas behind it,

The natural logarithm of e, [ln(e) or loge(e)], is 1, as e1 = e. ln(e) is like saying; e to the power of what equals e,

The natural logarithm of 1, [ln(1) or loge(1)] is 0, since e0 = 1. ln(1) is like saying, e to the power of what equals 1.

In words, think of it like:

ln(this) = what,

e to the power of this equals what

For example,

ln (ex) = x

e(ln(x)) = x

In the same way that 3x ÷ 3 = x and (x ÷ 3)(3) = x - multiplying by 3 and dividing by 3 are inverse operations to each other, in the same way that the natural logarithm and the exponential, e, are inverse operations to each other

So, if you were trying to solve an algebraic equation like,

12 = ex

Then apply the inverse to both sides of the equation, which would be the natural logarithm

ln (12) = ln(ex)

ln(12) = x

Using a calculator,

x = 2.484....

And for, Ordinary Logarithms, they are symbolized by loga(x) where loga is the inverse of ax.

So if, we had,

2x = 8,

Then, apply the inverse operation to both sides of the equation,

log2(2x) = log2(8)

x = log2(8) = 3,

Which makes sense, as 23 = 8!

Derivatives of Logarithms / Differentiating Logarithms

Definition - Logarithm Derivative

d over dx derivative operator (ln [x]) = 
1
x

and

d over dx derivative operator (ln [f(x)]) = 
f’(x)
f(x)

Beginner Derivatives of Natural Logarithms

Differentiate the following Natural Logarithms:

a)d over dx derivative operator (ln(2x))

First, look at our formula for derivatives of natural logarithms with a function involved.

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

So for d over dx derivative operator (ln(2x)) first identify what the function is in the natural logarithm.

In this case, f(x) = 2x. We will also need f'(x) = 2

Now sub these into our formula.

d over dx derivative operator
(ln(2x))  =   
2
2x
  =   
1
x

b)d over dx derivative operator (ln(3x2 + 1))

First recall our formula for derivatives of natural logarithms

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

Identify our function in d over dx derivative operator (ln(3x2 + 1))

In this case, f(x) = 3x2 + 1 which, when differentiated gives f'(x) = 6x

Now sub these into our formula.

d over dx derivative operator
(ln(3x2 + 1))  =   
6x
3x2 + 1

Derivatives of Natural Logarithms

View the solutions to the questions below to see examples of how to find the derivative of natural logarithmic functions.

The solution to finding the derivative of ln(x^3 + 2)to power of 2
The solution to finding the derivative of ln(cos(x))

Definition / recall - Change of Base Formula for Logarithms

A method we will use in our work on differentiating natural logarithms and logarithms will be the Change of Base Formula. We will be able to mathematically rewrite normal logarithms as natural logarithms, then be able to differentiate them.

loga(x) = 
logb(x)
logb(a)

In other words, for any logarithm of base a of x, we can rewrite it to equal logarithm of any positive base b of x divided by logarithm of any positive base b of a. So, if we pick that any positive base to be the exponential, e, we get,

loga(x) = 
loge(x)
loge(a)
 = 
ln (x)
ln (a)

[as loge(x) = ln (x)]

Change of Base Formula

View the examples below that show how to change logarithms using the Change of Base Formula (C.B.F.)

The solution to changing the base of log 27 to the base 3
The solution to changing the base of log 27 to the base pi.

Definition - Derivative of a Logarithm / Normal Logarithm

d over dx derivative operator loga(x) = 
1
(x)ln(a)
d over dx derivative operator loga(f(x)) = 
f'(x)
f(x)ln(a)

Differentiating Normal Logarithms

Follow the steps in the worked examples below to how to differentiate normal logarithms

The solution to differentiating  cos(x) to the base 3
The solution to differentiating  e^x to the base pi

Derivative of ax for any a ≥ 0

Definition - Derivative of ax for any a ≥ 0

d over dx derivative operator (ax) = (ax)(ln(a)) for any a > 0

Derivatives with more exponents

Follow the steps in the worked examples that show how to calculate derivatives

The solution to differentiating  3 to the power of x
The solution to differentiating  12 to the power of x

Example - Derivatives of a Number to the Power with Chain Rule

Follow the steps in the worked example below.

The solution to differentiating  4 to the power of sin(x)

Derivatives with exponents and Chain Rule

See below how to calculate derivatives with the aid of the Chain Rule

The solution to differentiating  5 to the power of x^3
The solution to differentiating  sin of (3 to the power of x)