Derivative of a product of functions

Premise

In calculus, more often than not, we have to differentiate functions which are a product of two or more functions.

In a mathematical format this looks like:

Latex formula

 

For example:

Latex formula

 

or sometimes a product of multiple functions:

Latex formula

 

For example:

Latex formula

 

which by a simple substitution can be turned into a product of 2 functions:

Latex formula

 

Solution

The derivative of a product of two functions is the following:

Latex formula

 

Derivation

We start from basic principles, the definition of a derivative:

Latex formula

 

Now, using the product property of limits, we have:

Latex formula

 

Which turns into:

Latex formula

 

that finally leads to the product rule:

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"

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