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Point)Use implicit differentiation with partial derivatives (as described in Example 4 page 732) to find dyldx where it is assumed that y is a differentiable functi...

Question

Point)Use implicit differentiation with partial derivatives (as described in Example 4 page 732) to find dyldx where it is assumed that y is a differentiable function of xxy+vxy =4dy dx(Note that your answers should be a function of x, y )

point) Use implicit differentiation with partial derivatives (as described in Example 4 page 732) to find dyldx where it is assumed that y is a differentiable function of x xy+vxy =4 dy dx (Note that your answers should be a function of x, y )



Answers

Implicit differentiation Use Theorem 15.9 to evaluate dy/dx. Assume each equation implicitly defines y as a differentiable function of $x$. $$\sqrt{x^{2}+2 x y+y^{4}}=3$$

In this problem, we will cover computing partial derivatives. So we want to first find the partial derivative of respect to X. Of this function. Yeah. And that means holding the variable. Why fixed? So that means we're going to treat the expression E to the way as a constant while trying to find the derivative of the only X. Term which is X. So I'm going to write it like this times E. Y. And the derivative of X is just one. So our partial derivative with respect to X is just going to be E to the Y. Now to find out where this partial derivative is going to be equal to zero. We would normally just set E to the y equal to zero. However, The function each to the Y Never Touches zero. If you were to graph it, It only gets closer to zero as the exponents rosen magnitude in the negative direction. And because he never touches zero, that means that there is no point in which. Well, first, the partial derivative with respect to X, is going to be zero and second, no point In which both partial derivatives are going to be equal to zero as well.

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In this problem, we will cover computing partial derivatives. So we want to first begin with the partial derivative with respect two X. And that means we are going to hold why fixed? So that means treating it like a constant. So when we're looking for the partial derivative of prospective X, we are just going to focus on the X term because taking the derivative of the white term is just going to be zero since it's a constant. So our X term is X squared and the derivative of that is going to be two X. Now we want to find out where this derivative is going to be equal to zero. And to do that, we just set two X equal 20 And we see that it is going to be zero whenever X is equal to zero. So now we want to find the partial derivative with respect to why. And to do that, we hold X now fixed instead of Y. So we're going to treat X like constant. And that means that the derivative with respect to why of that X term is just going to be zero. So we're only focusing on Y squared and the german invoice squared is just going to be two. Why? So now we want to find out where the partial derivative perspective, why is going to be equal to zero? So that means just setting to y equal to zero. And we see that this equals zero When Y. Equals zero. And this implies that the only point in which both partial derivatives are equal to zero is going to be at the origin, Which is the zero.

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