Differentiate a function f(x) = (x+1/x)²

Correct Answer is : 2(x−1x3)2(x-{1}{x³})2(x−x31)

Correct Answer is : 2(x−1x3)2(x-{1}{x³})2(x−x31)

Test Index

AP DSC School Assistant Math Model Test 1

Section: Subject Specific

Question : 114 of 160

Marks: +1, -0

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

Solution:

Concept:

To differentiate a function of the form

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

, first expand the square, then apply the power rule of differentiation:

\frac{d}{dx} x^n = n x^{n-1}

for each term, and the derivative of a constant is zero.

Explanation:

Step 1: Expand the given function:

f(x) = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = x^2 + 2 + x^{-2}

Step 2: Differentiate term by term with respect to

x

\frac{d}{dx}(x^2) = 2x

\frac{d}{dx}(2) = 0

\frac{d}{dx}(x^{-2}) = -2x^{-3} = -\frac{2}{x^3}

Step 3: Combine the results:

f'(x) = 2x - \frac{2}{x^3} = 2\left(x - \frac{1}{x^3}\right)

Answer:

2\left(x - \frac{1}{x^3}\right)

which corresponds to option C.

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