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Assam CEE Logarithmic — practice questions

Explore 3 free Logarithmic questions in Assam CEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Logarithmic questions and exercises with answers

Showing questions 1–3 of 3.

If $y = x^x$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = \log(\sin x)$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →If $y = (\sin x)^x$, then $\dfrac{dy}{dx}$ equals:Read the answer and worked solution →

Worked practice questions

If $y = x^x$, then $\dfrac{dy}{dx}$ equals:

  1. $x^x(1 + \ln x)$
  2. $x^{x-1}$
  3. $x^x \ln x$
  4. $x \cdot x^{x-1}$
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Correct answer: A. $x^x(1 + \ln x)$

Take $\ln$: $\ln y = x \ln x$. **Differentiate:** $\frac{1}{y}\frac{dy}{dx} = \ln x + 1$ ⇒ $dy/dx = y(1 + \ln x) = x^x(1 + \ln x)$. Classic logarithmic differentiation.

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If $y = \log(\sin x)$, then $\dfrac{dy}{dx}$ equals:

  1. $\dfrac{1}{\sin x}$
  2. $\cos x / \sin^2 x$
  3. $\cot x$
  4. $\tan x$
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Correct answer: C. $\cot x$

$dy/dx = \frac{1}{\sin x} \cdot \cos x = \cot x$.

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If $y = (\sin x)^x$, then $\dfrac{dy}{dx}$ equals:

  1. $(\sin x)^x [\sin x + x\cos x]$
  2. $x (\sin x)^{x-1} \cos x$
  3. $(\sin x)^x \cdot \cos x$
  4. $(\sin x)^x [\log(\sin x) + x \cot x]$
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Correct answer: D. $(\sin x)^x [\log(\sin x) + x \cot x]$

**Logarithmic diff:** $\ln y = x \ln(\sin x)$. **Differentiate:** $y'/y = \ln(\sin x) + x \cdot (\cos x/\sin x) = \ln(\sin x) + x\cot x$. **So $y' = y[\ln(\sin x) + x\cot x]$.**

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