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Chap3_Sec5

# Chap3_Sec5 - Example 3(3.5 The figure drawn with the...

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a. If x 2 + y 2 = 25, find . b. Find an equation of the tangent to the circle x 2 + y 2 = 25 at the point (3, 4). dy dx IMPLICIT DIFFERENTIATION Example 1 (3.5)

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The expression dy / dx = - x / y gives the derivative in terms of both x and y . It is correct no matter which function y is determined by the given equation. For instance, for , we have: However, for , we have: NOTE 2 ( ) 25 y f x x = = - 2 25 dy x x dx y x = - = - - 2 ( ) 25 y g x x = = - - 2 2 25 25 dy x x x dx y x x = - = - = - - -
a. Find y’ if x 3 + y 3 = 6 xy . b. Find the tangent to the folium of Descartes x 3 + y 3 = 6 xy at the point (3, 3). c. At what points in the first quadrant is the tangent line horizontal? IMPLICIT DIFFERENTIATION Example 2 (3.5)

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s A glance at the figure confirms that this is a reasonable value for the slope at (3, 3). s So, an equation of the tangent to the folium at (3, 3) is: y – 3 = – 1( x – 3) or x + y = 6. IMPLICIT DIFFERENTIATION The tangent is horizontal at (0, 0) and at (2 4/3 , 2 5/3 ), which is approximately (2.5198, 3.1748).
Find y’ if sin( x + y ) = y 2 cos x . IMPLICIT DIFFERENTIATION

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Unformatted text preview: Example 3 (3.5) The figure, drawn with the implicit-plotting command of a computer algebra system, shows part of the curve sin( x + y ) = y 2 cos x . s As a check on our calculation, notice that y’ = -1 when x = y = 0 and it appears that the slope is approximately -1 at the origin. IMPLICIT DIFFERENTIATION Example 3 Find y” if x 4 + y 4 = 16. IMPLICIT DIFFERENTIATION Example 4 (3.5) Recall the definition of the arcsine function: Differentiating sin y = x implicitly with respect to x , we obtain: 1 sin means sin and 2 2 y x y x y π-= =-≤ ≤ 1 cos 1 or cos dy dy y dx dx y = = DERIVATIVE OF ARCSINE FUNCTION Now, cos y ≥ 0, since – π /2 ≤ y ≤ π /2. So, Thus, 2 2 cos 1 sin 1 y y x =-=-2 1 2 1 1 cos 1 1 (sin ) 1 dy dx y x d x dx x-= =-=-DERIVATIVE OF ARCSINE FUNCTION Differentiate: a. b. f ( x ) = x arctan 1 1 sin y x-= x Example 5 (3.5) DERIVATIVES OF ITFs...
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Chap3_Sec5 - Example 3(3.5 The figure drawn with the...

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