If ^-1(x²+y²)=a², then find dy/dx.

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 21(b)

Set in the CBSE Class 12 board exam of 2025, 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 $tan^{- 1} (x^{2} + y^{2}) = a^{2}$ , then find $\frac{d y}{d x}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

The right-hand side $a^{2}$ is a constant.

Hint 2

Remove the inverse tangent by taking tan of both sides. The new right side is still a constant.

Hint 3

Differentiate implicitly, remembering that the derivative of a constant is 0.

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.

Take tan of both sides: $tan^{- 1} (x^{2} + y^{2}) = a^{2} \Rightarrow x^{2} + y^{2} = tan a^{2}$ , which is a constant.
½ mark
Differentiate both sides with respect to $x$ : $2 x + 2 y \frac{d y}{d x} = 0$ .
1 mark
Solve for the derivative: $\frac{d y}{d x} = - \frac{x}{y}$ .
½ mark

Answer: $\frac{d y}{d x} = - \frac{x}{y}$

Where the marks are

Rewrites the equation as $x^{2} + y^{2} = tan a^{2}$ (½ mark)
Differentiates to get $2 x + 2 y y^{'} = 0$ (1 mark)
Obtains $\frac{d y}{d x} = - \frac{x}{y}$ (½ 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 paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 21(b)

Hints

Step-by-step solution

Revise the chapter

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