Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation...
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Transcript of Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation...
![Page 1: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/1.jpg)
Implicit DifferentiationSection 3.7a
![Page 2: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/2.jpg)
Consider the equation:3 3 9 0x y xy
Is this a function?
Is the equationdifferentiable?
If so, how do wedifferentiate?
1y f x
2y f x
3y f x
![Page 3: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/3.jpg)
3 3 9 0x y xy
We use implicit differentiation, so namedb/c the functions are defined implicitly (hidden)within the original equation
1y f x
2y f x
3y f x
![Page 4: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/4.jpg)
Implicit Differentiation Process We treat y as a differentiable function of x!!!
1.Differentiate both sides of the equation withrespect to x.
2. Collect the terms with dy/dx on one side ofthe equation.
3. Factor out dy/dx.
4. Solve for dy/dx.
Because of the Chain Rule, any time differentiatingany term containing a y, also multiply by dy/dx!!!
![Page 5: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/5.jpg)
Initial Guided Practice
Find dy/dx: 2y x
2 1dyydx
1
2
dy
dx y
Does this answer make sense graphically?
Implicit Differentiation:
Solve for the derivative:
![Page 6: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/6.jpg)
Initial Guided Practice
Find the slope of the circle at2 2 25x y
2 2 0dy
x ydx
First, find the slope of any point on the circle viaimplicit differentiation:
dy x
dx y
3 3
4 4
dy
dx
Slope at the given point:
Again, verify thisanswer graphically!
3, 4
2 2dyy xdx
![Page 7: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/7.jpg)
Initial Guided PracticeShow that the slope dy/dx is defined at every point onthe graph of 22 siny x y
2 2 cosdy dy
x ydx dx
Imp Diff:
2 cos 2dy dy
y xdx dx
2 cos 2dy
y xdx
2
2 cos
dy x
dx y
This formula for dy/dxis defined at every point(x,y) except for thosepoints at which cos(y)=2…
Which never happens!!!
![Page 8: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/8.jpg)
We can use all of this new info to expand the POWERof the POWER RULE!!! A proof:
First, let p and q be integers with q > 0 and suppose that:
q py x
1
1
p
q
dy px
dx qy
1 1q pdyqy px
dx
1
1
p
qp q
p x
q x
pq p qy x x 1p
p p q
p x
q x
1p p p qpx
q
1p qpx
q
Imp. Diff.!
Subst. for y
Law of Exp.
Final Answer!
![Page 9: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/9.jpg)
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 thesquare root function???
1n ndx nx
dx
![Page 10: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/10.jpg)
Additional Guided PracticeFor each of the following, find dy/dx.
2 31 6y x
1 34 1 6x
1 321 6 6
3
dyx
dx
![Page 11: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/11.jpg)
Additional Guided PracticeFor each of the following, find dy/dx.
3 3 18x y xy
2 23 3 18 1dy dy
x y x ydx dx
2 23 18 18 3dy dy
y x y xdx dx
2 23 18 18 3dy
y x y xdx
![Page 12: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/12.jpg)
Additional Guided PracticeFor each of the following, find dy/dx.
3 3 18x y xy
2
2
6
6
y x
y x
2 23 18 18 3dy
y x y xdx
2
2
18 3
3 18
dy y x
dx y x
![Page 13: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/13.jpg)
Additional Guided PracticeFor each of the following, find dy/dx.
sinx y xy
1 cos 1dy dyy x ydx dx
cos 1dy dyy x ydx dx
cos 1dy
y x ydx
![Page 14: Implicit Differentiation Section 3.7a. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?](https://reader038.fdocuments.net/reader038/viewer/2022110400/56649dbd5503460f94aaf9e6/html5/thumbnails/14.jpg)
Additional Guided PracticeFor each of the following, find dy/dx.
sinx y xy
cos 1dy
y x ydx
1
cos
dy y
dx y x
1
cos
y
x y