The Product Rule

By:Mohammad-Ali Bandzar | Dec 25 2019

The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and states that the derivative of loading... will be loading.... This rule only works if f and g are both differentiable functions. This pattern works for any number of functions as i will demonstrate below.

With A Single Function

With Three Functions

Proof

To start off our proof we will write out the definition of the derivative:
we can now add and subtract loading... from the numerator which is allowed becuase by adding then subtracting the same thing we will not change our end result.
we will then factor loading... from the first and second last terms
we will then factor loading... from our remaining 2 terms in the numerator
Now if we recall the addition property of limits:
Now if we recall the multiplicative property of limits:
We can now simplify the left most limit, since loading... is differentiable and since all differentiable functions are continuous we can conclude that loading... must be continuous allowing us to sub what h is approaching directly into loading... giving us loading.... We can now substitute that into our equation:
we can then apply the multiplicitive property of limits again on the right limit to get:
since our limit variable h is not involved in the function in anyway we can simplify it further to:
now if we recall the definition of the derivitive:
we can substitute those in to our equaiton:
we have now completed our proof