George Khnouf Blog Post 4

Blog Post 4:

Good morning class! My name is professor George Khnouf and I will be substituting for Professor little who is out sick today.  Last class I was informed that you learned to find the derivatives of the different types of functions: Exponential Functions, Logarithmic Functions, Trigonometric functions, Power Functions, and Linear Functions.  Today I will be teaching you the different rules that you can apply these concepts to: The Chain Rule, the Product Rule and The Quotient Rule.

1- Chain Rule:

The Chain Rule is the rule we use in order to find the derivative of the composition of 2 functions.

- Chain Rule: F(g(t))’= F’(g(t)) x g’(t)

The easiest way to explain this rule and how we apply it is by giving you the following example rather than writing out and verbally listing.  Generally that is the best way to get students to understand mathematical procedures.

Imagine we have the following function:

f(x)= 3(2t³+10t²)³

The first thing we must do in this case is to identify the outer and inner functions and their derivatives:

- Outer Function:

F(t)= 3t³

Derivative of the Outer Function:

F(t)’= 9t²

- Inner Function:

g(t)= 2t³+10t²

Derivative of Inner Function:

g(t)’= 6t²+20t

After extracting the above information we substitute it accordingly into the equation: F(g(t))’= F’(g(t)) x g(t)’

F(t)’= 9(2t³+10t²)² x (6t²+20t)

And this is how you use the chain rule in order to find the derivative of the composition of 2 functions.

2- Product Rule:

This rule is used in order to find the derivatives of the product of 2 functions.

Product Rule: (F(t) x g(t))’= F(t)’ x g(t) + F(t) x g(t)’

In order to explain how this rule can be applied I will use another example rather than verbally write out the method and steps in sentences:

Imagine we had the following function:

h(t)= 3t² 7e^3t

Firstly, we must identify the 2 functions present within this function and find their derivatives:

Function 1:

F(t)= 3t²

F(t)’= 6t 

Function 2:

g(t)= 7e^3t

g(t)’= 21e^3t

Once this information has been found we can substitute our findings into the equation of the product rule:

(F(t).g(t))’= 6t x 7e^3t + 3t² x 21e^3t

This is how you use the product rule in order to find the derivative of the product of 2 functions.

3- Quotient Rule:

It is similar to the product rule, but it is different in terms of the equation of the rule itself and the fact that it finds the derivative of the quotient of 2 functions instead of their product:

Quotient Rule: (F(t)/g(t))’= (F(t)’ x g(t) – F(t) x g(t)’)/ g(t)²

In order to explain this rule I will be using the same example I used in in the explanation of the product rule:

  

 Imagine we had the following function:

h(t)= 3t² 7e^3t

Firstly, we must identify the 2 functions present within this function and find their derivatives:

Function 1:

F(t)= 3t²

F(t)’= 6t 

Function 2:

g(t)= 7e^3t

g(t)’= 21e^3t

Once this information has been found we can substitute our findings into the equation of the quotient rule:

(F(t)/g(t))’= (6t x 7e^3t -3t² x 21e^3t)/ (7e^3x)²

= (6t x 7e^3t -3t² x 21e^3t)/ 7e^6x

This is how we use the quotient rule in order to find the derivative of a quotient of 2 functions.

I hope my explanation was sufficient and as detailed as possible and I hope that now you understand and know how to use the chain, product and quotient rules.