Differentiate x^x with respect to x x.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2026: question 22(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section B, question 22(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22(a) · 2 marks · Very short answerContinuity and Differentiability

Differentiate $x^{x}$ with respect to $x lo g x$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Differentiating one function with respect to another means finding $\frac{d u / d x}{d v / d x}$ .

Hint 2

Use logarithmic differentiation for $x^{x}$ .

Hint 3

Use the product rule for $x lo g x$ , then divide the two derivatives.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let $u = x^{x}$ and $v = x lo g x$ . We need $\frac{d u}{d v}$ .
½ mark
$lo g u = x lo g x \Rightarrow \frac{1}{u} \frac{d u}{d x} = 1 + lo g x \Rightarrow \frac{d u}{d x} = x^{x} (1 + lo g x)$
½ mark
$\frac{d v}{d x} = x \cdot \frac{1}{x} + lo g x = 1 + lo g x$
½ mark
$\frac{d u}{d v} = \frac{d u / d x}{d v / d x} = \frac{x ^{x} ( 1 + l o g x )}{1 + l o g x} = x^{x}$
½ mark

Answer: $\frac{d ( x ^{x} )}{d ( x l o g x )} = x^{x}$

Where the marks are

Takes $u = x^{x}$ , $v = x lo g x$ (½ mark)
$\frac{d u}{d x} = x^{x} (1 + lo g x)$ (½ mark)
$\frac{d v}{d x} = 1 + lo g x$ (½ mark)
$\frac{d u}{d v} = x^{x}$ (½ mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Logarithmic Differentiation

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 22(a)

Hints

Step-by-step solution

Revise the chapter

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