Section 37a Consider the equation Is this a function Is the equation differentiable If so how do we differentiate We use implicit differentiation so named bc the functions are defined implicitly hidden ID: 544971
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Slide1
Implicit Differentiation
Section 3.7aSlide2
Consider the equation:
Is this a function?
Is the equation
differentiable?
If so,
how
do we
differentiate?Slide3
We use
implicit differentiation
, so named
b/c the functions are defined implicitly (hidden)
within the original equation Slide4
Implicit Differentiation Process
We treat
y
as a differentiable function of
x
!!!
Differentiate both sides of the equation with
respect to x.
2. Collect the terms with dy/dx on one side of the equation.
3. Factor out dy/dx.4. Solve for dy/dx.
Because of the Chain Rule, any time differentiating
a
ny term containing a
y, also multiply by dy
/dx
!!!Slide5
Initial Guided Practice
Find
dy
/dx
:
Does this answer make sense graphically?
Implicit Differentiation:
Solve for the derivative:Slide6
Initial Guided Practice
Find the slope of the
circle at
First, find the slope of
any
point on the circle via
implicit differentiation:
Slope at the given point:
Again, verify
this
answer
graphically
!Slide7
Initial Guided Practice
Show that the slope
dy
/
dx
is defined at every point on
the graph of
Imp Diff:
This formula for
dy
/
dx
is defined at every point
(
x
,
y
) except for thosepoints at which
cos(y)=2…
Which never happens!!!Slide8
We can use all of this new info to expand the POWER
of the POWER RULE!!!
A proof:
First, let
p
and
q be integers with
q > 0 and suppose that:
Imp. Diff.!
Subst. for y
Law of Exp.
Final Answer!Slide9
The EXPANDED Power Rule:
If
n
is any rational number, then
(If n < 1, then the derivative does not exist at x = 0)
WHY NOT???
Note: Before, the power rule worked for integers only…
now, it works for any rational number power (i.e., fraction)
What is the derivative of
the
square root function???Slide10
Additional Guided Practice
For each of the following, find
dy
/
dx
.Slide11
Additional Guided Practice
For each of the following, find
dy
/
dx
.Slide12
Additional Guided Practice
For each of the following, find
dy
/
dx
.Slide13
Additional Guided Practice
For each of the following, find
dy
/
dx
.Slide14
Additional Guided Practice
For each of the following, find
dy
/
dx
.