The Product Rule (Level 3) In this video we will go over problems that
require us to use the graphs to find the derivative of the given function.
Let’s go over the first example. Let p(x) = f(x)g(x). Find p’(1) by using
the graphs of f and g. Alright this example is similar to the problems
from the previous videos. We are given a function that is formed by the product of two distinct
functions in this case f(x) and g(x), we also don’t have an analytical expression for
the function but we are provided with the graphs of these two functions. For this problem
we are asked to find the derivative of p evaluated at x = 1. Let's first find an expression for
the derivative of p(x), since p(x) is formed by the product of f(x) and g(x) we go ahead
and apply the product rule. Applying the product rule we obtain the following expression for
the derivative. Next we need to evaluate this expression at x = 1. This means that we need
to find the value of f(1), f prime of 1, g(1) and g prime of 1. Unfortunately these
values are not provided to us like they were in the previous video. This time around we
need to figure out these values by making use of the graph of f(x) and g(x). To find
f(1) we go ahead and find the y coordinate of the red graph when x = 1, looking at the
graph we see that it would be equal to 6. In the same manner the y coordinate of the
blue graph at x = 1 will be equal to 4, this would correspond to the value of g(1). Now
to find g prime and f prime evaluated at x = 1 we need to remember the geometric interpretation
of the derivative. In this case the derivative represents the slope of the tangent line at
every given value of x of the function. This means that in order to evaluate g prime at
x = 1, we need to calculate the slope of the tangent line of g(x) at x = 1. Since the blue
graph at x = 1 is formed by a line we can find the slope by calculating the rise over
the run in this case it is going to be equal to negative ½ . Lastly f prime evaluated at
x = 1 is going to be equal to the slope of the tangent line of the red graph at x = 1, This
value can be calculated in the same manner as g prime of 1, in this case it is going
to be equal 1 over 1 which is just equal to 1. Having found all the required values we
go ahead and simplify the expression, obtaining the final answer equal to 1. This problem
is testing a couple of things, first it is testing to see if you understand the product
rule, second it is testing to see if you understand what g(1) and f(1) represent in this case
they are the just y coordinate on their respective graph at x = 1, lastly do you know how
to find g and f prime by using the geometric definition of the derivative. Remember it
is just the slope of the tangent line at any given point on the function.
Alright let’s try the final example. Let p(x) = f(x)g(x). Find p’(4) by using
the graphs of f and g. Similar to the last example let's go ahead
and find an expression for p prime of x. From the previous example we know that it is going
to be equal to the following expression after using the product rule. Next let's go ahead
and find p prime evaluated at x = 4. We first need to find what f(4) is equal to, looking
at the red graph we see that the y coordinate of this curve at x = 4 is equal to 1. Likewise
g(4) can be found by using the blue graph in this case it will be equal to 8. Next let’s
go ahead and find g prime and f prime evaluated at x = 4. Once again in order to find the value
of these expressions we need to calculate the slope of the tangent line at x = 4 for
each graph. G prime of 4 is going to be equal to 0 since the slope of the tangent line at
this point is a horizontal line with no slope or in this case 0 slope. Lastly f prime of
4 is going to be equal to positive 1/2. Substituting these values into the derivative and simplifying
the expression we obtain the final answer equal to 4.
Keep in mind that whenever you need to find the derivative of a function formed
by a product of functions you need to make use of the product rule.
In our next video we will learn how to deal with functions formed by a quotient of two
distinct functions.