Derivative of a function of functions (Chain Rule)

Premise

In calculus we often end up having to differentiate a function of functions

Latex formula

 

for example:

Latex formula

 

Solution

The solution is arrived at using what is known as the Chain Rule, which states to take the derivative of the outside function and multiply it by the derivative of the inside function:

Latex formula

 

Derivation

The proof is very easy, we start from basic principles and look first at the derivative of g(x):

Latex formula

 

Now, for the outer function:

Latex formula

 

Let’s make a variable substitution:

Latex formula

 

Following the same line of thinking as we did for the derivative of g:

Latex formula

 

Substituting the values back in, we have:

Latex formula

 

About Vincenzo Barbato

Known to his friends as Enzo, he's an outside-the-box engineer/researcher whose interests and expertise span many fields, including controls systems, multi-physics simulations, mechatronics, oil technologies, data analysis and machine vision just to name a few.

Refusing to grow up, he's on a continuous journey to develop simple and creative solutions that have the power of disrupting industries by optimizing systems and processes.

Married to a beautiful wife, with two beautiful daughters and two identical twin boys, his home is a never ending chaotic fountain of inspiration.

His outlook on life:

"Never blindly accept what you're told, listen, but then question, with curiosity, creativity and collaboration we can change the world"

About Enzo

Known to his friends as Enzo, he's an outside-the-box engineer/researcher whose interests and expertise span many fields, including controls systems, multi-physics simulations, mechatronics, oil technologies, data analysis and machine vision just to name a few. Refusing to grow up, he's on a continuous journey to develop simple and creative solutions that have the power of disrupting industries by optimizing systems and processes. Married to a beautiful wife, with two beautiful daughters and two identical twin boys, his home is a never ending chaotic fountain of inspiration. His outlook on life: "Never blindly accept what you're told, listen, but then question, with curiosity, creativity and collaboration we can change the world"

Leave a Reply

Your email address will not be published. Required fields are marked *