2014 implicit differentiation
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Calculus BC
2014 Implicit Differentiation
Implicit Differentiation
Equation for a line:
Explicit Form
<One variable given explicitly in terms of the other>
Implicit Form
<Function implied by the equation>
Differentiate the Explicit
< Explicit: , y is function of x >
Differentiation taking place with respect to x. The derivative is explicit also.
y mx b
Ax By C
24 3 4y x x
8 3dy xdx
Implicit Differentiation
Equation of circle:
To work explicitly; must work two equations
2 2 9y x
29y x
Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly.
( Usually non-functions)
29y x
Implicit Differentiation
Chain Rule Pretend y is some function like
so becomes
(A)
(B)
(C)
Note: Use the Leibniz form. Leads to Parametric and Related Rates.
2 2 3y x x 2 4( 2 3)x x 4y
Find the derivative with respect to x
< Assuming - y is a differentiable function of x >
32y
4y
2 3x y
Implicit Differentiation
(D) Product Rule
(E) Chain Rule 3( )xy
2xy
Implicit Differentiation
To find implicitly.
EX: Diff Both Sides of equation with respect to x
Solve for
dydx
2 2 9x y dydx
EX 1:3 2 25 4y y y x
(a) Find the derivative at the point ( 5, 3 ) , at ( -1,-3 )
(b) Find where the curve has a horizontal tangent.
(c) Find where the curve has vertical tangents.
Ex 2:
3 3 2x y xy
< Folium of Descartes >
Why Implicit?
3 3 2x y xy
< Folium of Descartes > Explicit Form:
3 6 3 3 6 33 31
1 1 1 18 82 4 2 4
y x x x x x x
3 6 3 3 6 33 32 1
1 1 1 1 13 8 82 2 4 2 4
y y x x x x x x
3 6 3 3 6 33 33 1
1 1 1 1 13 8 82 2 4 2 4
y y x x x x x x
Ex 2 Graph:
3 3 2x y xy < Folium of Descartes >
2
3 3
3 3, ; 11 1t tx y tt t
Plot the Folium of Descartes on your graphing calculator and determine the portion of the folium generated when
(a) t < -1 ; (b) -1 < t 0 ; (c) t > 0
Parametric Form:
2nd Derivatives
NOTICE:The second derivative is in terms of x , y , AND dy /dx.
The final step will be to substitute back the value of dy / dx into the second derivative.
EX: Our friendly circle. Find the 2nd Derivative.2 2 9x y
2nd DerivativesEX: Find the 2nd Derivative.
23 5xy
Higher DerivativesEX: Find the Third Derivative.
sin( )y x
Last update
• 10/19/10
p. 162 11 – 29 odd
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