The Quotient Rule (Level 3) In this video we will tackle on problems that
require us to use a set a values and graphs to find the derivative of a function. Let’s
first go over examples that require us to use a set of values to find the derivative.
If f of x equals h of x over x, where h of 2 equals 4 and h prime of 2 equals -3 find
f prime of 2. Alright here we are asked to find the derivative
of f evaluated at x = 2, let's first find the derivative of f(x) before we attempt to evaluate
it. Notice that f(x) is formed by the quotient of two functions in this case we have h(x)
and x, this means that we have to apply the quotient rule, applying the quotient rule
we obtain the following expression for the derivative. Note that we could also rewrite
this function with the denominator written as a negative exponent in this case we would
have to apply the product rule. For this example let’s continue on with the quotient rule.
Having found an expression for the derivative we go ahead and evaluate it at
x = 2, from the given values we know that h prime of 2 is equal to negative 3, and h
of 2 is equal to 4, substituting these expressions and simplifying we obtain negative 5 as our
final answer. Alright let’s try another example.
If f of x equals 1 over g of x, where g of 0 equals 2 and g prime of 0 equals 1, find
f prime of 0. Alright here we have a function that contains
the constant 1 in the numerator and the function g(x) in the denominator. Technically this
is still a quotient of two functions so we go ahead and find an expression for the derivative
by making use of the quotient rule. Applying the quotient rule we obtain the following,
notice that we could have also have identified this function as a reciprocal function and
made use of the reciprocal rule, regardless of the rule that you applied you should end
up with the following expression for the derivative negative g prime of x over g of x squared.
Next we need to find the derivative of f evaluated at x = 0, so we go ahead and substitute the
values provided to us, g prime of 0 is going to be equal to 1, and g of 0 is going to be
equal to 2, simplifying the expression we obtain the final value
equal to negative 1/4. Alright now let’s deal with a different
type of problem; let’s 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 q(x) equal f of x over g of x. Find q prime of 4 by using the graphs of f and g.
Alright in this example we are provided with the graph of f(x) and g(x) we need to use
these graphs to find the value of q prime evaluated at x = 4. Let’s first find an
expression for the derivative of q(x) before we start analyzing the graphs. Notice that
q(x) is formed by the quotient of f(x) and g(x) so we naturally need to apply the quotient
rule, applying the quotient rule we obtain the following expression for the derivative.
Next we need to evaluate this derivative at x = 4, in order to evaluate it we need values
for g(4), f(4) and their respective derivatives evaluated at x = 4. We will obtain these values
from the graph of f(x) and g(x), let's first find g(4) and f(4), these values are essentially
the y coordinate of the functions at x = 4, so looking at the graph we see that g(4) is
equal to 3, and f(4) is equal to 7. Now let’s find the values for the derivative of f(x)
and g(x), recall that the geometric interpretation of the derivative is represented by the slope
of the tangent line to the curve of the function at a particular value of x. This means that
f prime of 4 represents the slope of the tangent line to the curve of the function at x = 4,
we can calculate the slope of the tangent line at this point by calculating the rise
over the run in this case it is going to be equal to -1 over 1, which is just equal to
-1, in the same manner g prime of 4 can be calculated by finding the slope of the tangent
line on the curve g(x) at x = 4, looking at the graph of g(x) we see that the slope
at this point is going to be equal to 0 since the slope is flat at this point. Having found all the
required values we go ahead and substitute them into our expression for the derivative
evaluated at x = 4, simplifying the expression we obtain the final answer equal to negative
1/3. Alright make sure you understand the difference between a function evaluated at
a given value of x , and the derivative of a function evaluated at a given value of x,
the first represents the y-coordinate of the function being evaluated at the specific value
of x and the later represents the slope of the tangent line to the curve of the function
at that specific value of x. Alright let's go over the final example.
Let q of x equal f of x over g of x. Find q prime of 7 by using the graphs of f and
g. Ok similar to the previous example, we need
to find the derivative of this function evaluated at a specific value of x. Let's first find
the derivative of q(x), the derivative of this function is essentially going to be the
same as the previous example, it is just a matter of applying the quotient rule, applying
the quotient rule we obtain the following expression. Having found an expression for
the derivative we go ahead and evaluate it at x = 7, for this we need to find g(7), f(7),
g prime of 7 and f prime of 7, let's first find g(7) and f(7), these values represent
the y coordinate of each curve at x = 7, in this case g(7) is going to be equal to 4,
in the same manner f(7) is also going to be equal to 4 since the curve of f(x) and g(x)
intersect at exactly the same point. Now we need to find the derivative of f(x) at x = 7,
once again this is essentially going to be the slope of the tangent line of the red curve
at x = 7, we can find the rise over the run from two points on the curve to calculate
the slope of the tangent line, in this case the slope has a rise of 2 and run of 1,
in other words 2 over 1 or simply just 2. In the same manner the derivative of g(x) at x = 7
can be found in the same way, in this case, it is going to be equal to -1 over 1 or simply
-1, having found the values of the functions and their derivatives it is just a matter
of simplifying the expression, and reducing the fraction, doing that we obtain the final
answer equal to 3/4. Alright keep in mind that whenever you need
to find the derivative of a function formed by a quotient of functions you need to make
use of the quotient rule. In our next video we will learn how to deal with functions formed
by a composition of functions.