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Solve the given problems by using implicit differentiation.Show that two tangents to the curve $x^{2}+x y+y^{2}=7$ at the points where it crosses the $x$ -axis are parallel.

Calculus 1 / AB

Chapter 23

The Derivative

Section 8

Differentiation of Implicit Functions

Derivatives

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Lectures

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Find the two points where …

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Parallel tangents Find the…

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Find points on the curve $…

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Find equations of the tang…

05:13

Tangent lines Carry out th…

01:23

sketch the curve $f(x, y)=…

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Sketch the curve $f(x, y)=…

03:38

Tangents parallel to the c…

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Use implicit differentiati…

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right. We have the equation X squared, plus X y plus y squared equals seven, and they want us to find the points where this crosses the X axis. And so if it's crossing the X axis, that means why equals zero. And so if we plug in Y equals zero, obviously the X Y and the Y squared term become zero, so we have X squared equal seven or X equals plus or minus the square root of seven Now, next thing they asked us to do was to show that the tangent lines at those two points are parallel to each other. Well, parallel lines have the same slope, and the first derivative, when it's evaluated at a point, gives us the slope of the tangent line. So let's find the first derivative here. So driven about X squared is gonna be two x plus, doing implicit differentiation and using the product rule. We have X times, the directive of why we should be R D Y d Ekstrom and use the white prime notation, plus why times the derivative of X, which is just one and then plus the derivative of Y squared, will be to why times why prime and that'll equal zero. And so now we can solve this for why crime or we just plug in our values of plus and minus square root of seven as is. And that's what I'm gonna do. First, I'm gonna plug in X equals Positive Square root seven. So that would give us to square with seven plus the square root of seven. Why crying plus R Y value is zero. So why is zero? And that means this is also zero. So I don't even need my plus sign. I can just say that to square root of seven plus the square root of seven white prime equals zero And then we subtract the two square root of seven from both sides and we get square with a seven y prime equals negative to square of seven. And then when we divide, we get why Prime equals negative too. And now for X equals the negative square root of seven. Same thing putting that in. We have negative two square root of seven minus square to seven times by prime and our Y values were still zero, and that equals zero. So negative square with a seven. Why prime all equal, positive two square root of seven. So that means that why prime equals negative to when we divide by the negative square root of seven. So we have the same first derivative. So that means the changing lines have the same slope. So, yes, they are parallel to each other.

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