If y=x^ x+( x)^x; x>1, then evaluate dy/dx

dy/dx=(2 x)/(x)x^ x+( x)^x [( x)+(1)/( x) ]

MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 28(B)

From the official CBSE sample question paper for 2026-27, section C, question 28(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(B) · 3 marks · Short answerContinuity and Differentiability

If $y = x^{l o g x} + (lo g x)^{x}$ ; $x > 1$ , then evaluate $\frac{d y}{d x}$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Split $y$ into two functions $u$ and $v$ ; you cannot take the log of a sum directly.

Hint 2

Take the log of each part separately, differentiate, then add $\frac{d u}{d x}$ and $\frac{d v}{d x}$ .

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 $u = x^{l o g x}$ and $v = (lo g x)^{x}$ , so $y = u + v$ . Taking logs: $lo g u = (lo g x)^{2}$ and $lo g v = x lo g (lo g x)$ . Differentiating: $\frac{1}{u} \frac{d u}{d x} = \frac{2 l o g x}{x}$ and $\frac{1}{v} \frac{d v}{d x} = lo g (lo g x) + x \cdot \frac{1}{l o g x} \cdot \frac{1}{x} = lo g (lo g x) + \frac{1}{l o g x}$
1 mark
$\frac{d u}{d x} = x^{l o g x} \cdot \frac{2 l o g x}{x}$ and $\frac{d v}{d x} = (lo g x)^{x} (lo g (lo g x) + \frac{1}{l o g x})$
1 mark
$\frac{d y}{d x} = \frac{d u}{d x} + \frac{d v}{d x} = \frac{2 l o g x}{x} x^{l o g x} + (lo g x)^{x} (lo g (lo g x) + \frac{1}{l o g x})$
1 mark

Answer: $\frac{d y}{d x} = \frac{2 l o g x}{x} x^{l o g x} + (lo g x)^{x} [lo g (lo g x) + \frac{1}{l o g x}]$

Where the marks are

Takes logarithms of each part and differentiates (1 mark)
Gets $\frac{d u}{d x}$ and $\frac{d v}{d x}$ (1 mark)
Adds to obtain $\frac{d y}{d x}$ (1 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 paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 28(B)

Hints

Step-by-step solution

Revise the chapter

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