Given the function f(x)=(1-x)/(3x^(3)-4), find f^(')(x) in simplified form. Answer: f^(')(x)=

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Given the function  f(x)=(1-x)/(3x^(3)-4), find  f^(')(x) in simplified form. Answer:  f^(')(x)=

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Quotient Rule Explanation: To find the derivative of the function f(x) = (1-x)/(3x^3-4), we will use the quotient rule. The quotient rule states that if we have a function that is the quotient of two functions, u(x)/v(x), then its derivative f'(x) is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = 1-x and v(x) = 3x^3-4.Derivative of u(x): First, we need to find the derivative of u(x) = 1-x. The derivative of a constant is 0, and the derivative of -x with respect to x is -1. Therefore, u'(x) = 0 - 1 = -1.Derivative of v(x): Next, we need to find the derivative of v(x) = 3x^3-4. The derivative of 3x^3 with respect to x is 9x^2, and the derivative of a constant is 0. Therefore, v'(x) = 9x^2 - 0 = 9x^2.Applying Quotient Rule: Now we apply the quotient rule. f'(x) = (v(x)u'(x) - u(x)v'(x))/(v(x))^2. Plugging in the derivatives we found, we get f'(x) = ((3x^3-4)(-1) - (1-x)(9x^2))/((3x^3-4)^2).Simplify Numerator: Simplify the numerator of the derivative. f'(x) = (-3x^3 + 4 - 9x^2 + 9x^2x)/((3x^3-4)^2). Notice that -9x^2 + 9x^2x simplifies to -9x^2(1-x).Further Simplify Numerator: Further simplify the numerator. f'(x) = (-3x^3 + 4 - 9x^2 + 9x^3)/((3x^3-4)^2). Combine like terms in the numerator, which are -3x^3 and 9x^3.Combine Like Terms: After combining like terms, we get f'(x) = (6x^3 + 4 - 9x^2)/((3x^3-4)^2). This is the simplified form of the derivative.

Given the function $f(x)=\frac{1-x}{3 x^{3}-4}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function f(x)=1−x3x3−4 f(x)=\frac{1-x}{3 x^{3}-4} f(x)=3x3−41−x​, find f′(x) f^{\prime}(x) f′(x) in simplified form.\newlineAnswer: f′(x)= f^{\prime}(x)= f′(x)=

Given the function $f(x)=\frac{1-x}{3 x^{3}-4}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$