Given the function y=(2x^(3))/(2+x^(3)), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

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Given the function  y=(2x^(3))/(2+x^(3)), find  (dy)/(dx) in simplified form. Answer:  (dy)/(dx)=

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Identify Function: Identify the function that needs to be differentiated. The function given is y=(2x^(3))/(2+x^(3)). We need to find the derivative of this function with respect to x, which is denoted as (dy)/(dx).Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function y = u/v, where both u and v are functions of x, then the derivative of y with respect to x is given by (dy)/(dx) = (v(du)/(dx) - u(dv)/(dx))/(v^2). Here, u = 2x^3 and v = 2 + x^3.Differentiate u and v: Differentiate u and v with respect to x. The derivative of u = 2x^3 with respect to x is (du)/(dx) = 6x^2. The derivative of v = 2 + x^3 with respect to x is (dv)/(dx) = 3x^2.Substitute Derivatives: Substitute the derivatives into the quotient rule formula. Using the derivatives from Step 3, we substitute into the quotient rule formula: (dy)/(dx) = ((2 + x^3)(6x^2) - (2x^3)(3x^2)) / (2 + x^3)^2.Simplify Expression: Simplify the expression. (dy)/(dx) = (12x^2 + 6x^5 - 6x^5) / (4 + 4x^3 + x^6). This simplifies to (dy)/(dx) = (12x^2) / (4 + 4x^3 + x^6).Factor Out Common Terms: Further simplify the expression by factoring out common terms if possible. We can factor out a 4 from the numerator and denominator: (dy)/(dx) = (4 * 3x^2) / (4 * (1 + x^3 + (1/4)x^6)). This simplifies to (dy)/(dx) = (3x^2) / (1 + x^3 + (1/4)x^6).

Given the function $y=\frac{2 x^{3}}{2+x^{3}}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

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Given the function $y=\frac{2 x^{3}}{2+x^{3}}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$