If -y^(2)-y^(3)-5x^(3)=y then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

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If  -y^(2)-y^(3)-5x^(3)=y then find  (dy)/(dx) in terms of  x and  y. Answer:  (dy)/(dx)=

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Differentiate -y^(2): We are given the equation -y^(2) - y^(3) - 5x^(3) = y. To find (dy)/(dx), we need to differentiate both sides of the equation with respect to x, treating y as a function of x (implicit differentiation).Differentiate -y^(3): Differentiate the term -y^(2) with respect to x. Since y is a function of x, we use the chain rule: the derivative of y^(2) with respect to y is 2y, and then we multiply by (dy)/(dx) (the derivative of y with respect to x). The differentiation gives us -2y(dy)/(dx).Differentiate -5x^(3): Differentiate the term -y^(3) with respect to x. Using the chain rule again, the derivative of y^(3) with respect to y is 3y^(2), and then we multiply by (dy)/(dx). The differentiation gives us -3y^(2)(dy)/(dx).Differentiate y: Differentiate the term -5x^(3) with respect to x. Since x is the variable we are differentiating with respect to, we simply use the power rule: the derivative of x^(3) is 3x^(2). The differentiation gives us -15x^(2).Combine and set equal: Differentiate the term y with respect to x. Since y is a function of x, the derivative is simply (dy)/(dx). The differentiation gives us (dy)/(dx).Isolate (dy)/(dx): Combine all the differentiated terms and set the equation equal to the derivative of the right side of the original equation, which is 0 since the derivative of a constant is 0. So, we have -2y(dy)/(dx) - 3y^(2)(dy)/(dx) - 15x^(2) = (dy)/(dx).Factor out (dy)/(dx): We need to solve for (dy)/(dx). To do this, we isolate terms containing (dy)/(dx) on one side of the equation. This gives us (dy)/(dx) - 2y(dy)/(dx) - 3y^(2)(dy)/(dx) = 15x^(2).Divide to solve for (dy)/(dx): Factor out (dy)/(dx) from the left side of the equation. This gives us (dy)/(dx)(1 - 2y - 3y^(2)) = 15x^(2).Divide both sides of the equation by (1 - 2y - 3y^(2)) to solve for (dy)/(dx). This gives us (dy)/(dx) = 15x^(2) / (1 - 2y - 3y^(2)).

If $-y^{2}-y^{3}-5 x^{3}=y$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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If $-y^{2}-y^{3}-5 x^{3}=y$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$