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MHTCET 15 May 2024 Shift 2 Paper

Question : 103 of 150

Marks: +1, -0

y=\frac{x^{\frac{2}{3}}-x^{\frac{-1}{3}}}{x^{\frac{2}{3}}+x^{\frac{-1}{3}}}, x \neq 0

(x+1)^{2} y_{1}=

Solution:

Concept:

Simplify the given function before differentiating to make the derivative easy to compute.

Explanation:

Given:

y = \frac{x^{\frac{2}{3}} - x^{-\frac{1}{3}}}{x^{\frac{2}{3}} + x^{-\frac{1}{3}}}

Factor out

x^{-\frac{1}{3}}

from both numerator and denominator:

y = \frac{x^{-\frac{1}{3}}(x - 1)}{x^{-\frac{1}{3}}(x + 1)}

Cancel the common factor:

y = \frac{x - 1}{x + 1}

Differentiate using the quotient rule:

y_1 = \frac{(x+1)(1) - (x-1)(1)}{(x+1)^2}

Simplify the numerator:

y_1 = \frac{2}{(x+1)^2}

Multiply both sides by

(x+1)^2

(x+1)^2 y_1 = 2

Answer:

(x+1)^2 y_1 = 2

Correct option: A. 2

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