Implicit Differentiation and Function Notation
General Contents
Implicit Differentiation Contents
Index
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Find
for
Equation 1
given that
f
(2) = 3 .
How do we interpret the symbols
? Is multiplication involved?
No, this group of symbols describes a function named “
f
” which depends on the variable
x
.
How do we interpret
?
It describes the value of the function “
f
” when the independent variable
x
has the value 2.
We frequently use the variable “
y
” for our dependent variable. What is the dependent variable here?
The function
is the dependent variable here.
How do we interpret
?
It represents the derivative of . In the analysis below, we’ll use the more familiar equivalent,
.
How do we interpret
?
It represents, the derivative of
, evaluated at
x
= 2.
How do we proceed to solve the problem?
We can differentiate
Equation 1
implicitly and substitute values for
x
and
. Then we can solve for
and substitute
x
= 2.
In differentiating the first term, what rules of differentiation apply?
Since it is primarily a product, we need the Product Rule.
How do we handle the exponents?
With the Power Rule.
Set up the differentiation of the product in the first term.
What will the new exponent of
be after differentiation.
Just as with the differentiation of any quantity to the fourth power, the new exponent will be 3.
Complete the differentiation.
We get
Equation 2
Why is the
factor there?
It is the result of applying the Chain Rule.
Differentiate the second term in
Equation 1
We get
Equation 3
Combine
Equations2 and 3
to get the derivative of
Equation 1
.
Replace
Substitute
x
= 2.
We get
Substitute
f
(2) = 3.
We get
Simplify.
Solve for
.
Finally, we get
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General Contents
Implicit Differentiation Contents
Index