JAMIA 2025 — Mathematics PYQ

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Question

JAMIA 2025 — Mathematics PYQ

JAMIA | Mathematics | 2025

The derivative of $x^{x}$ is:

Choose the correct answer:

Explanation

Mathematical Solution:

Maan lijiye:

y = x^x

Step 1: Dono taraf log (natural log) lene par:

\ln y = \ln(x^x)

Logarithm ki property $(\ln a^b = b \ln a)$ ka upyog karke:

\ln y = x \ln x

Step 2: Dono taraf $x$ ke saapeksh (with respect to $x$ ) differentiation karne par:

\frac{d}{dx}(\ln y) = \frac{d}{dx}(x \ln x)

Yahan RHS mein Product Rule $(\frac{d}{dx}[uv] = u \frac{dv}{dx} + v \frac{du}{dx})$ lagega:

\frac{1}{y} \frac{dy}{dx} = x \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}(x)

\frac{1}{y} \frac{dy}{dx} = x \cdot \left(\frac{1}{x}\right) + \ln x \cdot (1)

\frac{1}{y} \frac{dy}{dx} = 1 + \ln x

Step 3: $\frac{dy}{dx}$ nikalne ke liye $y$ ko doosri taraf le jayein:

\frac{dy}{dx} = y(1 + \ln x)

Ab $y$ ki jagah $x^x$ rakhne par:

\frac{dy}{dx} = x^x(1 + \ln x)

Final Answer:

The derivative of $x^x$ is:

\frac{d}{dx}(x^x) = x^x(1 + \ln x)

Choose the correct answer:

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