If y = √(√(x)), prove that √(x) dy/dx = (1+y^4)/(4y).

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 21(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section B, question 21(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(b) · 2 marks · Very short answerContinuity and Differentiability

If $y = tan x$ , prove that $x \frac{d y}{d x} = \frac{1 + y ^{4}}{4 y}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

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

Hint 1

Apply the chain rule in three layers: square root, tan, then $x$ .

Hint 2

Multiply by $x$ and write $sec^{2} x$ as $1 + tan^{2} x$ .

Hint 3

Note that $tan x = y^{2}$ .

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.

Differentiate $y = tan x$ by the chain rule: $\frac{d y}{d x} = \frac{1}{2 tan x} \cdot sec^{2} x \cdot \frac{1}{2 x}$
1 mark
So $x \frac{d y}{d x} = \frac{sec ^{2} x}{4 tan x} = \frac{1 + tan ^{2} x}{4 tan x}$ . Since $y^{2} = tan x$ , $tan^{2} x = y^{4}$ and $tan x = y$ , giving $x \frac{d y}{d x} = \frac{1 + y ^{4}}{4 y}$ . Hence proved.
1 mark

Answer: $x \frac{d y}{d x} = \frac{1 + y ^{4}}{4 y}$ (proved)

Where the marks are

Correct chain-rule derivative of $tan x$ (1 mark)
Uses $sec^{2} = 1 + tan^{2}$ and $y^{2} = tan x$ to reach the result (1 mark)

Revise the chapter

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

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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 21(b)

Hints

Step-by-step solution

Revise the chapter

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