Given the function f(x)=(2-x^(3))/(2+5x^(3)), find f^(')(x) in simplified form. Answer: f^(')(x)=

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Given the function  f(x)=(2-x^(3))/(2+5x^(3)), find  f^(')(x) in simplified form. Answer:  f^(')(x)=

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Given function: Given the function f(x) = (2 - x^3) / (2 + 5x^3), we need to find its derivative f'(x). We will use the quotient rule for derivatives, which states that if we have a function g(x) = u(x) / v(x), then g'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2. Here, u(x) = 2 - x^3 and v(x) = 2 + 5x^3.Derivative of u(x): First, we find the derivative of u(x) = 2 - x^3. The derivative of a constant is 0, and the derivative of x^3 with respect to x is 3x^2. Therefore, u'(x) = 0 - 3x^2 = -3x^2.Derivative of v(x): Next, we find the derivative of v(x) = 2 + 5x^3. Similarly, the derivative of a constant is 0, and the derivative of 5x^3 with respect to x is 15x^2. Therefore, v'(x) = 0 + 15x^2 = 15x^2.Apply quotient rule: Now we apply the quotient rule. We have u'(x) = -3x^2 and v'(x) = 15x^2. Plugging these into the quotient rule formula, we get: f'(x) = ((-3x^2)(2 + 5x^3) - (2 - x^3)(15x^2)) / (2 + 5x^3)^2.Simplify numerator: We simplify the numerator of f'(x) by distributing the terms: (-3x^2)(2 + 5x^3) = -6x^2 - 15x^5, (2 - x^3)(15x^2) = 30x^2 - 15x^5. So, the numerator becomes: (-6x^2 - 15x^5) - (30x^2 - 15x^5) = -6x^2 - 15x^5 - 30x^2 + 15x^5.Combine like terms: We combine like terms in the numerator: -6x^2 - 15x^5 - 30x^2 + 15x^5 = -6x^2 - 30x^2 = -36x^2.Final derivative: Now we have the simplified form of the derivative: f'(x) = -36x^2 / (2 + 5x^3)^2.

Given the function $f(x)=\frac{2-x^{3}}{2+5 x^{3}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function $f(x)=\frac{2-x^{3}}{2+5 x^{3}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$