Q1468: If $-2x^{2}-5xy+y^{3}=76$, then find $\frac{dy}{dx}$....

Class CBSE Class 12 Mathematics Derivatives Q #1468

KNOWLEDGE BASED

UNDERSTAND

2 Marks 2025 AISSCE(Board Exam) VSA

If $-2x^{2}-5xy+y^{3}=76$, then find $\frac{dy}{dx}$.

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AI Tutor Explanation

Detailed Solution

Step 1: Differentiate both sides with respect to $x$

We are given the equation $-2x^{2}-5xy+y^{3}=76$. We need to find $\frac{dy}{dx}$. To do this, we will differentiate both sides of the equation with respect to $x$.

Step 2: Apply the differentiation rules

Differentiating $-2x^2$ with respect to $x$ gives $-4x$. Differentiating $-5xy$ with respect to $x$ requires the product rule: $\frac{d}{dx}(-5xy) = -5(x\frac{dy}{dx} + y\frac{d}{dx}(x)) = -5(x\frac{dy}{dx} + y(1)) = -5x\frac{dy}{dx} - 5y$. Differentiating $y^3$ with respect to $x$ requires the chain rule: $\frac{d}{dx}(y^3) = 3y^2\frac{dy}{dx}$. Differentiating $76$ with respect to $x$ gives $0$ since it is a constant.

Step 3: Combine the differentiated terms

Combining all the differentiated terms, we get: $$-4x - 5x\frac{dy}{dx} - 5y + 3y^2\frac{dy}{dx} = 0$$

Step 4: Isolate $\frac{dy}{dx}$ terms

Rearrange the equation to isolate the terms containing $\frac{dy}{dx}$: $$3y^2\frac{dy}{dx} - 5x\frac{dy}{dx} = 4x + 5y$$

Step 5: Factor out $\frac{dy}{dx}$

Factor out $\frac{dy}{dx}$ from the left side: $$\frac{dy}{dx}(3y^2 - 5x) = 4x + 5y$$

Step 6: Solve for $\frac{dy}{dx}$

Divide both sides by $(3y^2 - 5x)$ to solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{4x + 5y}{3y^2 - 5x}$$

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

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of implicit differentiation and apply the chain rule and product rule correctly to find the derivative.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure (implicit differentiation) to solve the problem. This involves knowing the steps and rules for differentiating implicit functions.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of implicit differentiation, a standard topic in the syllabus. The question is a direct application of a textbook method.