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CUET Mathematics 2025 PYQs2 June Shift 1. Free, no login required.

Calculus

past year questions, CUET Mathematics 2025 2 June Shift 1

Continuity & Differentiability

past year questions, Calculus, CUET Mathematics 2025 2 June Shift 1

Medium

common

If y=11−x33y = \frac{1}{\sqrt[3]{1-x^3}} then dydx\frac{dy}{dx} is equal to

Solution

✅ Correct Option: 2

Given: y=11−x33y = \frac{1}{\sqrt[3]{1-x^3}}

Rewriting in a simpler form for differentiation:

y=1(1−x3)1/3y = \frac{1}{(1-x^3)^{1/3}}

y=(1−x3)−1/3y = (1-x^3)^{-1/3}

Applying the chain rule with power n=−13n = -\frac{1}{3} and inner function f(x)=1−x3f(x) = 1-x^3:

dydx=−13⋅(1−x3)−13−1⋅ddx(1−x3)\frac{dy}{dx} = -\frac{1}{3} \cdot (1-x^3)^{-\frac{1}{3}-1} \cdot \frac{d}{dx}(1-x^3)

Simplifying the exponent:

−13−1=−43-\frac{1}{3} - 1 = -\frac{4}{3}

dydx=−13⋅(1−x3)−4/3⋅ddx(1−x3)\frac{dy}{dx} = -\frac{1}{3} \cdot (1-x^3)^{-4/3} \cdot \frac{d}{dx}(1-x^3)

The derivative of the inner function:

ddx(1−x3)=−3x2\frac{d}{dx}(1-x^3) = -3x^2

Substituting:

dydx=−13⋅(1−x3)−4/3⋅(−3x2)\frac{dy}{dx} = -\frac{1}{3} \cdot (1-x^3)^{-4/3} \cdot (-3x^2)

Simplifying the constants:

dydx=−13×(−3x2)×(1−x3)−4/3\frac{dy}{dx} = -\frac{1}{3} \times (-3x^2) \times (1-x^3)^{-4/3}

dydx=x2(1−x3)−4/3\frac{dy}{dx} = x^2(1-x^3)^{-4/3}

Therefore, dydx=x2(1−x3)−4/3\frac{dy}{dx} = x^2(1-x^3)^{-4/3}

Related questions:

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