If y= ((π)/(4)+x/2 ), then prove that dy/dx- x=0

MathsCBSE Class 12 · Sample paper, session 2025-26 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 22

From the official CBSE sample question paper for 2025-26, section B, question 22, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22 · 2 marks · Very short answerContinuity and Differentiability

If $y = lo g tan (\frac{π}{4} + \frac{x}{2})$ , then prove that $\frac{d y}{d x} - sec x = 0$

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

Use $\frac{d}{d x} lo g u = \frac{1}{u} \frac{d u}{d x}$ and $\frac{d}{d x} tan u = sec^{2} u \frac{d u}{d x}$ .

Hint 2

Write $tan$ and $sec^{2}$ in sines and cosines; look for $2 sin A cos A$ .

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.

Differentiating w.r.t. $x$ by the chain rule: $\frac{d y}{d x} = \frac{1}{tan ( \frac{π}{4} + \frac{x}{2} )} \cdot sec^{2} (\frac{π}{4} + \frac{x}{2}) \cdot \frac{1}{2}$ .
½ mark
$= \frac{cos ( \frac{π}{4} + \frac{x}{2} )}{sin ( \frac{π}{4} + \frac{x}{2} )} \cdot \frac{1}{cos ^{2} ( \frac{π}{4} + \frac{x}{2} )} \cdot \frac{1}{2} = \frac{1}{2 sin ( \frac{π}{4} + \frac{x}{2} ) cos ( \frac{π}{4} + \frac{x}{2} )} = \frac{1}{sin ( \frac{π}{2} + x )} = \frac{1}{cos x}$ [ $2 sin A cos A = sin 2 A$ ].
1 mark
So $\frac{d y}{d x} = sec x$ , i.e. $\frac{d y}{d x} - sec x = 0$ . Hence proved.
½ mark

Answer: $\frac{d y}{d x} = sec x$ , so $\frac{d y}{d x} - sec x = 0$

Where the marks are

Correct chain-rule derivative (½ mark)
Simplifies using $2 sin A cos A = sin 2 A$ to $1/ cos x$ (1 mark)
Concludes $\frac{d y}{d x} - sec x = 0$ (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 paperSample paper 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 22

Hints

Step-by-step solution

Revise the chapter

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