If 5y+y^(3)-y^(2)-x^(3)=0 then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

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If  5y+y^(3)-y^(2)-x^(3)=0 then find  (dy)/(dx) in terms of  x and  y. Answer:  (dy)/(dx)=

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Given Equation: We are given the equation 5y + y^3 - y^2 - x^3 = 0 and we need to find the derivative of y with respect to x, which is (dy/dx). To do this, we will use implicit differentiation, which involves taking the derivative of both sides of the equation with respect to x, while treating y as a function of x.Implicit Differentiation: First, we differentiate each term of the equation with respect to x. For the terms involving y, we will apply the chain rule, which states that the derivative of a function of a function is the derivative of the outer function times the derivative of the inner function. In this case, the ＂inner function＂ is y, which is a function of x, so we multiply by dy/dx after differentiating with respect to y.Differentiating 5y: Differentiating 5y with respect to x gives us 5(dy/dx), since the derivative of y with respect to x is dy/dx.Differentiating y^3: Differentiating y^3 with respect to x gives us 3y^2(dy/dx), by applying the chain rule (the derivative of y^3 with respect to y is 3y^2, and then we multiply by dy/dx).Differentiating -y^2: Differentiating -y^2 with respect to x gives us -2y(dy/dx), by applying the chain rule (the derivative of y^2 with respect to y is 2y, and then we multiply by dy/dx).Differentiating -x^3: Differentiating -x^3 with respect to x gives us -3x^2, since x is the variable we are differentiating with respect to.Combining Differentiated Terms: Now we combine all the differentiated terms to rewrite the equation: 5(dy/dx) + 3y^2(dy/dx) - 2y(dy/dx) - 3x^2 = 0.Factoring out dy/dx: We can factor out dy/dx from the terms that contain it: (dy/dx)(5 + 3y^2 - 2y) - 3x^2 = 0.Isolating dy/dx: Now we isolate dy/dx by adding 3x^2 to both sides and then dividing by (5 + 3y^2 - 2y): dy/dx = 3x^2 / (5 + 3y^2 - 2y).

If $5 y+y^{3}-y^{2}-x^{3}=0$ 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 $5 y+y^{3}-y^{2}-x^{3}=0$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$