If x^y=e^x-y prove that dy/dx=( x)/(((xe))²) and hence find its value at x=e.

dy/dx=( x)/(((xe))²); value at x=e is 14

MathsCBSE Class 12 · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 26(A)

From the official CBSE sample question paper for 2025-26, section C, question 26(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(A) · 3 marks · Short answerContinuity and Differentiability

If $x^{y} = e^{x - y}$ prove that $\frac{d y}{d x} = \frac{l o g x}{( l o g ( x e ) ) ^{2}}$ and hence find its value at $x = e$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Take logarithm of both sides and use $lo g e = 1$ .

Hint 2

Make $y$ the subject before differentiating, and write $1 = lo g e$ at the end.

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.

Taking log: $y lo g x = (x - y) lo g e = x - y$ [ $lo g e = 1$ ] $\Rightarrow y (1 + lo g x) = x \Rightarrow y = \frac{x}{1 + lo g x}$ .
1 mark
By quotient rule: $\frac{d y}{d x} = \frac{( 1 + lo g x ) \cdot 1 - x \cdot \frac{1}{x}}{( 1 + lo g x ) ^{2}} = \frac{lo g x}{( 1 + lo g x ) ^{2}} = \frac{lo g x}{( lo g e + lo g x ) ^{2}} = \frac{lo g x}{( lo g ( x e ) ) ^{2}}$ . Hence proved.
1 mark
At $x = e$ : $\frac{d y}{d x} = \frac{lo g e}{( lo g e ^{2} ) ^{2}} = \frac{1}{( 2 ) ^{2}} = \frac{1}{4}$ .
1 mark

Answer: $\frac{d y}{d x} = \frac{l o g x}{( l o g ( x e ) ) ^{2}}$ ; value at $x = e$ is $\frac{1}{4}$

Where the marks are

Takes logs and expresses $y = \frac{x}{1 + l o g x}$ (1 mark)
Differentiates and shows the required form (1 mark)
Evaluates $\frac{1}{4}$ at $x = e$ (1 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Implicit 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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 26(A)

Hints

Step-by-step solution

Revise the chapter

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