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 just one function say loading... we can arbitarly define a second function loading... and multiply the two together to create a new function we will call loading...can you see how loading... is the same function as loading...? Now if we take the derivative of loading... we get loading... now if we recall our constant rule, we know that the derivative of a constant in this case 1 will be zero. So if we sub in zero we get loading... now if we sub in loading... we get loading... we have now shown that we can use our product rule for derivatives on a single function by defining a second function equal to one.

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

loading... Now, you maybe thinking that we should let loading... and loading... but that is not allowed. Because by the definition of the product rule both functions must be differentials and loading... is not a differentiable function. so we cannot directly apply the product rule here. what we would have to do is create a piecewise function where it becomes loading... when loading... or loading... when loading... which as a piecewise would look like this: loading... then we can differentiate the two segments independently and use the definition of the derivitive to find the derivitve at their point of intersection.