The Chain Rule Part 1

General Explanation and Base Work for the Chain Rule

The Chain Rule is how we combine functions together and use those combinations to give us results. We can use the Chain Rule to Differentiate functions that we otherwise would not have the method to differentiate.

Definition - The Chain Rule (in Leibniz Notation)

If y is a differentiable function of u, say y = f(u) and u is a differentiable function of x, say u = f(x) then y is a differentiable function of x.

dy
dx
=
dy
du
.
du
dx

In other words, we can get to the differentiable value of x from y, by going via u. u is connected to y and connected to x. Think of it like,

Yvonne knows Umar,

and Umar knows Xavier,

so Yvonne knows Xavier through Umar

One other point we can note that may help us understand why the Chain Rule works is by thinking of it like,

5
7
=
5
1
.
1
7

This can also be mathematically written as follows:

5
7
=
5
1
.
1
7
.
du
du

Think of the "du" as an irrelevance since, just like the 1, it will cancel out.

=
5
du
.
du
7

It still equals the same thing, however, mathematically it has use to us.

Now think of it like:

dy
dx
=
dy
du
.
du
dx

It may seem pointless to add in "du" as it can cancel out. However we will see this can be used to help us in differentiation.

The Chain Rule makes more sense once it is seen in an example.

Example - Beginner Leibniz Chain Rule

In this example, we will use Leibniz Chain Rule to differentiate an equation that can also be done simply by multiplying out the brackets and then differentiating. The purpose of this example is to introduce you to Leibniz Chain Rule. We will use the Chain Rule when it is not possible to differentiate using standard methods.

Use the Leibniz Chain Rule to differentiate:

y = 2(2x + 3)2

Substituting u = 2x + 3

We can differentiate by expanding the brackets but for the purpose of practicing the Leibniz Rule will do the substitution.

So first we substitute u = 2x + 3

y = 2(2x + 3)2

Becomes:

y = 2(u)2, y = 2u2

So now we have an equation with y & u and u = 2x + 3, u & x

When we differentiate y = 2u2 and u = 2x + 3

We get dy over du derivative operator = (2)2u2-1 = 4u & du over dx derivative operator = 2

Recall Leibniz Chain Rule:

dy
dx
=
dy
du
.
du
dx

So:

Now think of it like:

dy
dx
= 4u.2 = 8u

Now, lastly, substitute u = 2x + 3 back in.

dy
dx
= 8(2x + 3) = 16x + 24

If you like, try differentiating the original equation.

y = 2(2x + 3)2 = 2(4x2 + 12x + 9)

= 8x2 + 24x + 18

dy over dx derivative operator = 16x + 24

Trigonometric with Chain Rule

Follow the steps that use the Chain Rule to answer each of the two questions below.

question requiring finding the derivative of cos(3x^2 + 3)
question requiring finding the derivative of sin(4x + e^x)
part 1 of the solution to finding the derivative of cos(3x^2 + 3)
part 2 of the solution to finding the derivative of sin(4x + e^x)
More Chain Rule Examples

The Chain Rule - Composition Form

Definition - The Chain Rule - (in Composition form)

If we have a function, g(x) that is differentiable at x, and f is a function differentiable at g(x),

Then the composite f of g is differentiable at x and therefore,

(f of g)ā€™(x) =D( f( g(x) ) ) = fā€™( g(x) )( gā€™(x) )

As in the Leibniz Chain Rule, the first example will be one to familiarize yourself with the Composition Chain Rule.

Example - Beginner Composition Chain Rule

Use the Composite Chain Rule to differentiate:

y = 6square rooot symbol
x2 + x

First, recall the Composite Chain Rule:

D(f(g(x))) = f'(g(x)).g'(x)

So we must identify an inner and outer function.

For,   
y = 6square rooot symbol
x2 + x

Try to see where there are two functions.

y = 6square rooot symbol in green
x2 + x
We have the outer function,   
y = 6square rooot symbol in green
(something)
, and the inner function, x2 + x
We can write these as, f(x) = 6
square rooot symbol in green
x
, and g(x) = x2 + x

Now we calculate their derivatives.

f(x) = 6(x)½

f'(x) = (½)6(x)½ - 1

f'(x) = 3x

  =   
3
square rooot symbol in greenx

g(x) = 2x + 1

Now sub these into:

D(f(g(x))) = f'(g(x)).g'(x)

  =   
3(2x + 1)
square rooot symbol in green(x2 + x)
  =   
6x + 3
square rooot symbol(x2 + x)
More Chain Rule Examples

The Chain Rule with Product Rule

Follow the steps in the two worked examples below to see the use of the Product Rule and Chain Rule to find derivatives of functions.

Question asking to find the derivative of cos(x) times e^(2x)
Question asking to find the derivative of sin(2xe^x)
The solution to finding the derivative of cos(x) times e^(2x)
The solution to finding the derivative of sin(2xe^x)

The Table of Derivatives and The Chain Rule

The Chain Rule with arctan

The two examples below show the steps to differentiate functions with arctan using the Chain Rule.

Question asking to find the derivative of arctan(e^x)
Question asking to find the derivative of of arctan(3x)
The solution to finding the derivative of of arctan(e^x) using the Leibniz Chain Rule
The solution to finding the derivative of sin(3x) using the Composite Chain Rule

The Chain Rule with arcsin

The two examples below show the steps to differentiate functions with arctan using the Chain Rule.

Question asking to find the derivative of arcsin(x)
Question asking to find the derivative of of arcsin(e^x)
The solution to finding the derivative of of arcsin(x) using the Composite Chain Rule
The solution to finding the derivative of arcsin(e^x) using the Leibniz Chain Rule

The Chain Rule with sinh, cosh and ln

Follow the steps in the two worked examples below to see the Chain Rule used with sinh, cosh, and ln.

Question asking to find the derivative of asinh(2x + 5)
Question asking to find the derivative of of ln(1 - e^(3x))
The solution to finding the derivative of of asinh(2x + 5) using the Leibniz Chain Rule
The solution to finding the derivative of ln(1 - e^(3x)) using the Composite Chain Rule