Given the function y=(2x^(3)+1)/(3+x), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

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Given the function  y=(2x^(3)+1)/(3+x), 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)+1)/(3+x), and 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 + 1 and v = 3 + x.Differentiate u: Differentiate u with respect to x. The derivative of u = 2x^3 + 1 with respect to x is (du)/(dx) = 6x^2.Differentiate v: Differentiate v with respect to x. The derivative of v = 3 + x with respect to x is (dv)/(dx) = 1.Apply Quotient Rule: Apply the quotient rule using the derivatives from steps 3 and 4. Substitute the derivatives into the quotient rule formula to get (dy)/(dx) = ((3 + x)(6x^2) - (2x^3 + 1)(1))/((3 + x)^2).Simplify Expression: Simplify the expression. Expand the numerator to get (dy)/(dx) = (18x^2 + 6x^2) - (2x^3 + 1) and simplify further.Combine Like Terms: Combine like terms in the numerator. Combine the terms to get (dy)/(dx) = 24x^2 - 2x^3 - 1.Factor Numerator: Factor the numerator if possible. In this case, the numerator cannot be factored in a way that will cancel with the denominator, so we leave the expression as is.Final Simplified Form: Write the final simplified form of the derivative. The final simplified form of the derivative is (dy)/(dx) = (24x^2 - 2x^3 - 1)/((3 + x)^2).

Given the function $y=\frac{2 x^{3}+1}{3+x}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

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Given the function $y=\frac{2 x^{3}+1}{3+x}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$