If -2x² - 5xy + y³ = 76, then find dy/dx.

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 23(b)

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section B, question 23(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(b) · 2 marks · Very short answerContinuity and Differentiability

If $- 2 x^{2} - 5 x y + y^{3} = 76$ , 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 65/7/1.

Hints

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

Hint 1

Differentiate each term with respect to x, treating y as a function of x.

Hint 2

$\frac{d}{d x} (x y) = y + x \frac{d y}{d x}$ and $\frac{d}{d x} (y^{3}) = 3 y^{2} \frac{d y}{d x}$ .

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.

Differentiate both sides with respect to x, using the product rule on $x y$ : $- 4 x - 5 (y + x \frac{d y}{d x}) + 3 y^{2} \frac{d y}{d x} = 0$ , i.e. $- 4 x - 5 y - 5 x \frac{d y}{d x} + 3 y^{2} \frac{d y}{d x} = 0$ .
1½ marks
Collect the $\frac{d y}{d x}$ terms: $\frac{d y}{d x} (3 y^{2} - 5 x) = 4 x + 5 y$ , so $\frac{d y}{d x} = \frac{4 x + 5 y}{3 y ^{2} - 5 x}$ .
½ mark

Answer: $\frac{d y}{d x} = \frac{4 x + 5 y}{3 y ^{2} - 5 x}$

Where the marks are

Correct implicit differentiation, including the product rule for $x y$ (1½ marks)
Rearranges to get $\frac{d y}{d x} = \frac{4 x + 5 y}{3 y ^{2} - 5 x}$ (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 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 23(b)

Hints

Step-by-step solution

Revise the chapter

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