Mark Sanders
Dec 1, 2014
Applied Calculus
Professor Little
Hello,
my name is professor Sanders and I am going to explain the product and quotient
rules for derivatives.
When
you are trying to find the derivative of a function which involves variables
being multiplied together or divided into each other, you cannot simply use the
power rule. Using the power rule will mean that your answer will not be
relative to the function. You can, however, find the derivative through using
the product rule or the quotient rule.
Let’s
start with the product rule:
To
find the derivative of {F(x)*G(x)}, you must use the product rule.
The Product Rule
The
product rule id used to find the derivative of a function where two or more
variables are being multiplied together.
The product rule is:
F(x)’
* G(x) + F(x)* G(x)’= {F(x)*G(x)}’
In
words, this can be explained as multiplying the derivative of the first
variable times the second variable plus the first variable times the derivative
of the second variable.
Example:
Find
the derivative of the following function: F(x)= (3x^3)(5x^2)
Step One: Take the derivative of
the first term and multiply it by the second term.
(3x^3)*d/dx=
9x^2
(9x^2)(5x^2)
Step Two: Take the derivative of
the second term and multiply it by the first term.
(5x^2)*d/dx=
10x
(10x)(3x^3)
Step Three: Add the two answers
together to get the derivative of F(x)=(3x^x)(5x^2). This is the final answer
and is the derivative for the function.
(9x^2)(5x^2)+
(10x)(3x^3) = The derivative of F(x)=(3x^x)(5x^2).
The Quotient Rule
The
quotient rule is used when you are trying to find the derivative of a function
with a variable divided by another variable. The quotient rule is very similar
to the product rule. There are, however, two major differences. For one, you
don’t add, but rather subtract the results you get in the first two steps from
above. You also divide by the dividing term squared.
The quotient rule:
{F(x)’
* G(x) - F(x) * G(x)’} / G(x)^2= {F(x)/G(x)}’
In
words, you would multiply the derivative of the first term by the second term
and then subtract this by the derivative of the second term times the first
term. This all would then be divided the second term squared.
Example:
Find
the derivative of the following function: F(x)= (10x)/(2x^2)
Step One: Find the derivative of
the first term and multiply it by the second term.
(10x)d/dx=
10
(10)(2x^2)
Step Two: Find the derivative of
the second term and multiply it by the first term.
(2x^2)d/dx=
4x
(10x)(4x)
Step Three: Subtract the first
answer from step one by the answer from step two.
(10)(2x^2)
– (10x)(4x)
Step Four: Divide the answer from
step three by the second derivative squared. This is the final answer and is
the derivative of F(x)= (10x)/(2x^2).
{(10)(2x^2)
– (10x)(4x)}/(2x^2)^2= The derivative of
F(x)= (10x)/(2x^2).
Hi Mark I liked how you numbered the process because thats the way I teach myself things. I also like how you enforced the concepts with examples
ReplyDeletemark,
ReplyDeletelovely work explaining the derivative rules for products and quotients. other than a couple of typos, your calculations and explanations were good!
professor little