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Evaluate dy/dx at the given points.$$x y^{2}+3 x^{2}-y^{2}+15=0 ; \quad(-1,3)$$

$\left.\frac{d y}{d x}\right|_{(-1,3)}=\frac{1}{4}$

Calculus 1 / AB

Chapter 23

The Derivative

Section 8

Differentiation of Implicit Functions

Derivatives

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in this problem we're gonna use implicit differentiation to find the derivative or dy dx of this equation and then evaluated at the point negative 13 So to start we're just going to take the derivative of this function. So for X Y squared, if we're taking that derivative we need product rule which is going to be the first times the derivative of the second. So the derivative of Y squared would be to y. Dy dx Then plus the derivative the first which is just one times the second. So that would just stay wide squared. Then we go on. We need the derivative of three X squared which is just gonna be six X. The derivative of negative Y squared is negative two. Y. Do you? I D. X. The derivative of 15 0 Derivative is of zero is also zero. So our next step is to get dy dx on one side, everything else on the other. So I'm gonna go ahead and subtract this, Y squared and six X To the other side. So this will really become two X. Y. Do I. D X minus two. Y. Do you I. D. X equals negative six X minus Y squared. And since I want dy dx by itself I'm gonna factor that out. So I have to I. D. X. And then two Xy minus two. Y. That's going to equal the negative six X minus Y squared. So our last step here would just be to divide By two Xy -2 Y. So that would cancel out. And that would give us do I. D. X. So do I. D. X. Equals negative six X minus y squared over two X. Y minus two. Y. But this problem wants us to evaluate it at the 0.-13. So X. Is gonna be negative one. Why is going to be three? So everywhere there's an X. We placed with the negative one everywhere there's a why we're gonna replace with the three? So do I. D. X. Is gonna be negative six? The X. becomes a negative one. The Y. Becomes a three so minus three squared over two times negative one times three -2 times three. Which means now we just need to simplify this. So do I. D. X. Is gonna equal -6 times one is gonna be six -3 skirts. That's -9 over negative six minus six. So then we get DY dx is going to equal negative three over negative 12. So it's going to simplify the equal negative I mean positive 1/4.

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