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Find the f(x)=(2x-3)/(x^(3)+3x)

1313) Find the\newlinef(x)=2x−3x3+3x f(x)=\frac{2 x-3}{x^{3}+3 x}

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Q. 1313) Find the\newlinef(x)=2x−3x3+3x f(x)=\frac{2 x-3}{x^{3}+3 x}
  1. Identify Function & Need: Identify the function and the need to use the quotient rule since the function is a ratio of two functions.\newlineReasoning: f(x)=2x−3x3+3xf(x) = \frac{2x-3}{x^3 + 3x} is a quotient of two functions, numerator u(x)=2x−3u(x) = 2x-3 and denominator v(x)=x3+3xv(x) = x^3 + 3x.\newlineCalculation: No calculations in this step.
  2. Differentiate Numerator & Denominator: Differentiate the numerator u(x)=2x−3u(x) = 2x - 3 and the denominator v(x)=x3+3xv(x) = x^3 + 3x. Reasoning: Use basic differentiation rules. Calculation: u′(x)=2u'(x) = 2, v′(x)=3x2+3v'(x) = 3x^2 + 3.
  3. Apply Quotient Rule: Apply the quotient rule: (v(x)u′(x)−u(x)v′(x))/(v(x))2(v(x)u'(x) - u(x)v'(x)) / (v(x))^2.\newlineReasoning: Quotient rule formula is (vu′−uv′)/v2(vu' - uv') / v^2.\newlineCalculation: f′(x)=((x3+3x)(2)−(2x−3)(3x2+3))/(x3+3x)2f'(x) = ((x^3 + 3x)(2) - (2x-3)(3x^2 + 3)) / (x^3 + 3x)^2.
  4. Simplify Derivative Expression: Simplify the derivative expression.\newlineReasoning: Expand and simplify the terms in the numerator.\newlineCalculation: f′(x)=2x3+6x−6x3−9x+9(x3+3x)2=−4x3−3x+9(x3+3x)2f'(x) = \frac{2x^3 + 6x - 6x^3 - 9x + 9}{(x^3 + 3x)^2} = \frac{-4x^3 - 3x + 9}{(x^3 + 3x)^2}.

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