Question:
Assertion (A):
\[
\frac{d}{dx}\left(\frac{x^2\sin x}{\log x}\right)
=
\frac{x^2\sin x}{\log x}
\left(\cot x+\frac{2}{x}-\frac{1}{x\log x}\right)
\]
Reason (R):
\[
\frac{d}{dx}\left(\frac{uv}{w}\right)
=
\frac{uv}{w}
\left[
\frac{u'}{u}+\frac{v'}{v}-\frac{w'}{w}
\right]
\]
Assertion (A):
\[
\frac{d}{dx}\left(\frac{x^2\sin x}{\log x}\right)
=
\frac{x^2\sin x}{\log x}
\left(\cot x+\frac{2}{x}-\frac{1}{x\log x}\right)
\]
Reason (R):
\[
\frac{d}{dx}\left(\frac{uv}{w}\right)
=
\frac{uv}{w}
\left[
\frac{u'}{u}+\frac{v'}{v}-\frac{w'}{w}
\right]
\]
Show Hint
For functions involving products, quotients, and powers, logarithmic differentiation often makes the calculation easier and faster.
Updated On: Jun 22, 2026
- A is true, R is true and R is correct explanation of A
- A is true, R is true and R is not correct explanation of A
- A is true, R is not correct
- A is not correct, R is correct
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Identify the given function.
Let \[ y=\frac{x^2\sin x}{\log x} \] Here, \[ u=x^2,\qquad v=\sin x,\qquad w=\log x \]
Step 2: Use logarithmic differentiation.
Taking logarithm on both sides, \[ \log y=\log x^2+\log(\sin x)-\log(\log x) \] Differentiate with respect to \(x\): \[ \frac{y'}{y}=\frac{2}{x}+\cot x-\frac{1}{x\log x} \] Therefore, \[ y'=y\left(\frac{2}{x}+\cot x-\frac{1}{x\log x}\right) \] Substituting \[ y=\frac{x^2\sin x}{\log x}, \] we get \[ \frac{d}{dx}\left(\frac{x^2\sin x}{\log x}\right) = \frac{x^2\sin x}{\log x} \left(\cot x+\frac{2}{x}-\frac{1}{x\log x}\right) \] So, Assertion (A) is true.
Step 3: Check the reason.
The formula \[ \frac{d}{dx}\left(\frac{uv}{w}\right) = \frac{uv}{w} \left[ \frac{u'}{u}+\frac{v'}{v}-\frac{w'}{w} \right] \] is the logarithmic differentiation formula for a product divided by another function.
Hence, Reason (R) is also true.
Step 4: Check whether R explains A.
Since Assertion (A) is obtained directly by applying the formula given in Reason (R), Reason (R) is the correct explanation of Assertion (A).
Step 5: Final conclusion.
Therefore, \[ \boxed{\text{A is true, R is true and R is correct explanation of A}} \]
Let \[ y=\frac{x^2\sin x}{\log x} \] Here, \[ u=x^2,\qquad v=\sin x,\qquad w=\log x \]
Step 2: Use logarithmic differentiation.
Taking logarithm on both sides, \[ \log y=\log x^2+\log(\sin x)-\log(\log x) \] Differentiate with respect to \(x\): \[ \frac{y'}{y}=\frac{2}{x}+\cot x-\frac{1}{x\log x} \] Therefore, \[ y'=y\left(\frac{2}{x}+\cot x-\frac{1}{x\log x}\right) \] Substituting \[ y=\frac{x^2\sin x}{\log x}, \] we get \[ \frac{d}{dx}\left(\frac{x^2\sin x}{\log x}\right) = \frac{x^2\sin x}{\log x} \left(\cot x+\frac{2}{x}-\frac{1}{x\log x}\right) \] So, Assertion (A) is true.
Step 3: Check the reason.
The formula \[ \frac{d}{dx}\left(\frac{uv}{w}\right) = \frac{uv}{w} \left[ \frac{u'}{u}+\frac{v'}{v}-\frac{w'}{w} \right] \] is the logarithmic differentiation formula for a product divided by another function.
Hence, Reason (R) is also true.
Step 4: Check whether R explains A.
Since Assertion (A) is obtained directly by applying the formula given in Reason (R), Reason (R) is the correct explanation of Assertion (A).
Step 5: Final conclusion.
Therefore, \[ \boxed{\text{A is true, R is true and R is correct explanation of A}} \]
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