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

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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Write a formula for an arithmetic sequence

Full solution

Q. 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}=

Identify Function: Identify the function that needs to be differentiated. $\newline$ The function given is $y=\frac{2x^{3}}{2+x^{3}}$ . We need to find the derivative of this function with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function $y = \frac{u}{v}$ , where both $u$ and $v$ are functions of $x$ , then the derivative of $y$ with respect to $x$ is given by $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ . 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 $\frac{du}{dx} = 6x^2$ . The derivative of $v = 2 + x^3$ with respect to $x$ is $v$ $0$ .
Substitute Derivatives: Substitute the derivatives into the quotient rule formula. $\newline$ Using the derivatives from Step $3$ , we substitute into the quotient rule formula: $\newline$ $\frac{dy}{dx} = \frac{(2 + x^3)(6x^2) - (2x^3)(3x^2)}{(2 + x^3)^2}$ .
Simplify Expression: Simplify the expression. $\newline$ $(\frac{dy}{dx}) = \frac{12x^2 + 6x^5 - 6x^5}{4 + 4x^3 + x^6}$ . $\newline$ This simplifies to $(\frac{dy}{dx}) = \frac{12x^2}{4 + 4x^3 + x^6}$ .
Factor Out Common Terms: Further simplify the expression by factoring out common terms if possible. $\newline$ We can factor out a $4$ from the numerator and denominator: $\newline$ $\frac{dy}{dx} = \frac{4 \times 3x^2}{4 \times (1 + x^3 + (\frac{1}{4})x^6)}$ . $\newline$ This simplifies to $\frac{dy}{dx} = \frac{3x^2}{1 + x^3 + (\frac{1}{4})x^6}$ .

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