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

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

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Identify Function: Identify the function that needs to be differentiated. We are given the function y=(2x^(3)-3)/(3+2x^(3)). We need to find its derivative 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 - 3 and v = 3 + 2x^3.Differentiate u: Differentiate u with respect to x. The derivative of u = 2x^3 - 3 with respect to x is (du)/(dx) = 6x^2.Differentiate v: Differentiate v with respect to x. The derivative of v = 3 + 2x^3 with respect to x is (dv)/(dx) = 6x^2.Apply Quotient Rule: Apply the quotient rule using the derivatives from steps 3 and 4. Substitute (du)/(dx) and (dv)/(dx) into the quotient rule formula to get (dy)/(dx) = ((3 + 2x^3)(6x^2) - (2x^3 - 3)(6x^2))/((3 + 2x^3)^2).Expand Numerator: Expand the numerator of the derivative. Expanding the numerator, we get (dy)/(dx) = (18x^2 + 12x^5 - 12x^5 + 18x^2)/((3 + 2x^3)^2).Simplify Numerator: Simplify the numerator. Notice that the terms +12x^5 and -12x^5 cancel each other out. So, we have (dy)/(dx) = (18x^2 + 18x^2)/((3 + 2x^3)^2).Combine Like Terms: Combine like terms in the numerator. Combining like terms, we get (dy)/(dx) = 36x^2/((3 + 2x^3)^2).

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

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