Question

What is the value of ( frac{d y}{d x} ) at the point ( P(1,0) ) for the curve ( y=y(x) ) given by [ cos left(y^{2} ight)+y^{3}+x^{2}+4 x y=2 ? ] (a) -4 (b) -2 (c) -1 (d) ( -frac{1}{2} ) (e) ( quad-frac{1}{4} )

          What is the value of ( frac{d y}{d x} ) at the point ( P(1,0) ) for the curve ( y=y(x) ) given by
[
cos left(y^{2}
ight)+y^{3}+x^{2}+4 x y=2 ?
]
(a) -4
(b) -2
(c) -1
(d) ( -frac{1}{2} )
(e) ( quad-frac{1}{4} )
        
What is the value of ( fracd yd x ) at the point ( P(1,0) ) for the curve ( y=y(x) ) given by
[
cos left(y^2
ight)+y^3+x^2+4 x y=2 ?
]
(a) -4
(b) -2
(c) -1
(d) ( -frac12 )
(e) ( quad-frac14 )

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Calculus: Graphical, Numerical, Algebraic
Calculus: Graphical, Numerical, Algebraic
Ross L. Finney, Franklin D. Demana, Bet… 5th Edition
Chapter 4
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Transcript

-
00:01 In this problem, we are given an implicit equation relating y and x.
00:06 We want to find the value of dy over dx at the point x equal to 1 and y equal to 0.
00:13 So what we're going to do here is we're going to differentiate implicitly with respect to x.
00:21 So on the left hand side, we will have the derivative of cos of y squared plus y cubed plus x squared plus 4 times xy.
00:36 And on the right hand side, we will have the derivative of the constant 2.
00:43 And as we know, the derivative of all constants are equal to 0.
00:48 So this right hand side will simply be equal to 0.
00:50 So let's develop this left hand side, being careful to utilize chain rule when necessary.
00:56 So first, let's differentiate cos of y squared.
01:00 The derivative of cos is minus sine.
01:05 And now we differentiate the interior derivative of x squared, giving us 2y.
01:13 And let's not forget dy over dx due to chain rule...
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