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Section 2.5: Implicit Differentiation
1. Explicit Function
Most functions given have been written in the form
We can say
form.
is an explicit function of
For instance,
function of .
( ).
or the function is written in explicit
is a function where
is explicitly written as a
Finding a derivative of a function written in explicit form requires use of our
known differentiation rules. For the above function
2. Implicit Form
If a function cannot be written in explicit form, then it may be defined implicitly.
The equation
say is an implicit function of .
can not be written as
( ), therefore we
To take the derivative of a function written implicitly we require use of the chain
rule. This process is known as implicit differentiation.
Steps:
1.
2.
3.
4.
Remember that when you solve for
unknown function of .
you really found
( ) where
is an
To check your solution, substitute into the implicit definition and show the lefthand and right-hand sides of the equation are equivalent.
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Ex.) Find
1.
2.
3.
by implicit differentiation for each
3
Ex.) Find
by implicit differentiation then evaluate the derivative at the indicated point
at (
1.
2. (
)
(
)
)
at (
)
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3. Logarithmic Differentiation
On occasion is it convenient to use logarithms in differentiating non-logarithmic
function by the process of logarithmic differentiation.
Consider
√
.
To take the derivative we would need to
(1)
(2)
Instead, let’s apply a logarithm to each side and use laws of logarithms to simplify
the problem.
√