NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

Find $\frac{d}{dx} \left( \sqrt{x} - \frac{5}{\sqrt{x}} \right)$

Choose the correct answer:

Explanation

Step 1: Rewrite the expression in exponent form

We can rewrite the square roots as powers:

$\sqrt{x} = x^{1/2}$
$\frac{5}{\sqrt{x}} = 5x^{-1/2}$

So, the expression becomes:

\frac{d}{dx} (x^{1/2} - 5x^{-1/2})

Step 2: Differentiate term by term

Apply the power rule to each part:

For $x^{1/2}$ :

$\frac{d}{dx} (x^{1/2}) = \frac{1}{2} x^{(1/2 - 1)} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$
For $-5x^{-1/2}$ :

$\frac{d}{dx} (-5x^{-1/2}) = -5 \cdot \left( -\frac{1}{2} \right) x^{(-1/2 - 1)} = \frac{5}{2} x^{-3/2}$

Step 3: Combine the results

\frac{d}{dx} \left( \sqrt{x} - \frac{5}{\sqrt{x}} \right) = \frac{1}{2\sqrt{x}} + \frac{5}{2} x^{-3/2}

Conclusion:

Comparing this result with the given options, we find it matches option (a).

Correct Option:

(a) $\frac{1}{2\sqrt{x}} + \frac{5}{2} x^{-3/2}$