Question

question
given the function ( f(x)=\frac{x^{2}+5}{5 + 3x^{2}} ), find ( f^{prime}(x) ) in simplified form.

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if \(f(x)=\frac{u(x)}{v(x)}\), then \(f^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v(x)^{2}}\). Here, \(u(x)=x^{2}+5\), so \(u^{\prime}(x) = 2x\); \(v(x)=5 + 3x^{2}\), so \(v^{\prime}(x)=6x\).

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Step2: Expand the numerator

Expand \((2x)(5 + 3x^{2})=10x+6x^{3}\) and \((x^{2}+5)(6x)=6x^{3}+30x\).

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Step3: Simplify the numerator

Combine like - terms in the numerator: \(10x+6x^{3}-6x^{3}-30x=(10x-30x)+(6x^{3}-6x^{3})=-20x\)

Answer:

\(f^{\prime}(x)=\frac{-20x}{(5 + 3x^{2})^{2}}\)