Question:
If \(y = tan^{-1}⎛⎝\frac{log(\frac{e}{x^3})}{logex^3}⎞⎠+tan^{-1}⎛⎝\frac{log(e^4x^3)}{log(\frac{e}{x^{12}})}⎞⎠\), \(x\in (e^{-\frac{1}{3}},e^{\frac{1}{12}})\) then \(\frac{dy}{dx}\) is equal to...
If \(y = tan^{-1}⎛⎝\frac{log(\frac{e}{x^3})}{logex^3}⎞⎠+tan^{-1}⎛⎝\frac{log(e^4x^3)}{log(\frac{e}{x^{12}})}⎞⎠\), \(x\in (e^{-\frac{1}{3}},e^{\frac{1}{12}})\) then \(\frac{dy}{dx}\) is equal to...
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Write each logarithm in terms of u = 3 log x and use the tan inverse addition identities.
Updated On: Oct 1, 2026
- \(1\)
- \(0\)
- \(-1\)
- \(\frac{1}{e}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
Let \(u = \log x^3 = 3\log x\) (natural log, so \(\log e = 1\)). Then \(\log\frac{e}{x^3} = 1 - u\) and \(\log(ex^3) = 1 + u\).
Step 2: Key Formula or Approach:
\(\tan^{-1}\frac{1 - u}{1 + u} = \frac\pi4 - \tan^{-1}u\) and \(\tan^{-1}\frac{a + b}{1 - ab} = \tan^{-1}a + \tan^{-1}b\).
Step 3: Detailed Explanation:
First term: \(\tan^{-1}\frac{1 - u}{1 + u} = \frac\pi4 - \tan^{-1}u\).
Second term: \(\log(e^4x^3) = 4 + u\) and \(\log\frac{e}{x^{12}} = 1 - 4u\), so the fraction is \(\frac{4 + u}{1 - 4u}\), which gives \(\tan^{-1}4 + \tan^{-1}u\) for the given range.
\[ y = \frac\pi4 - \tan^{-1}u + \tan^{-1}4 + \tan^{-1}u = \frac\pi4 + \tan^{-1}4 \]
This is a constant, so \(\frac{dy}{dx} = 0\).
Final Answer:
\(\frac{dy}{dx} = 0\), option (B). \[ \boxed{0} \]
Let \(u = \log x^3 = 3\log x\) (natural log, so \(\log e = 1\)). Then \(\log\frac{e}{x^3} = 1 - u\) and \(\log(ex^3) = 1 + u\).
Step 2: Key Formula or Approach:
\(\tan^{-1}\frac{1 - u}{1 + u} = \frac\pi4 - \tan^{-1}u\) and \(\tan^{-1}\frac{a + b}{1 - ab} = \tan^{-1}a + \tan^{-1}b\).
Step 3: Detailed Explanation:
First term: \(\tan^{-1}\frac{1 - u}{1 + u} = \frac\pi4 - \tan^{-1}u\).
Second term: \(\log(e^4x^3) = 4 + u\) and \(\log\frac{e}{x^{12}} = 1 - 4u\), so the fraction is \(\frac{4 + u}{1 - 4u}\), which gives \(\tan^{-1}4 + \tan^{-1}u\) for the given range.
\[ y = \frac\pi4 - \tan^{-1}u + \tan^{-1}4 + \tan^{-1}u = \frac\pi4 + \tan^{-1}4 \]
This is a constant, so \(\frac{dy}{dx} = 0\).
Final Answer:
\(\frac{dy}{dx} = 0\), option (B). \[ \boxed{0} \]
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