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JEE MATHEMATICS MCQ: DIFFERENTIATION

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Let f(x)f(x) and g(x)g(x) be two functions having finite non-zero 3rd3 r d order derivatives f′′′(x)f^{\prime \prime \prime}(x) and g′′′(x)g^{\prime \prime \prime}(x) for all x∈Rx \in R. If f(x)g(x)=1f(x) g(x)=1 for all x∈Rx \in R, then f′′′f′−g′′′g′\frac{f^{\prime \prime \prime}}{f^{\prime}}-\frac{g^{\prime \prime \prime}}{g^{\prime}} is equal to
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Correct option: B
3(f′′f−g′′g)3\left(\frac{f^{\prime \prime}}{f}-\frac{g^{\prime \prime}}{g}\right)
Option B (3(f′′f−g′′g\frac{f^{\prime\prime}}{f}-\frac{g^{\prime\prime}}{g})) is correct because after differentiating the equation f(x)g(x) = 1 three times and rearranging, the expression simplifies to 3(f′′f−g′′g\frac{f^{\prime\prime}}{f}-\frac{g^{\prime\prime}}{g}).

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