2.3 Q-2

Question Statement

Differentiate the following expression with respect to $x$: $x^{-3} + 2x^{\frac{3}{2}} + 3$

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Background and Explanation

To solve this problem, we need to apply the power rule for differentiation. The power rule states that if you have a term of the form $x^n$, its derivative is $n \cdot x^{n-1}$. This rule can be applied even when $n$ is negative or fractional.

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Solution

Let the given function be: $Y = x^{-3} + 2x^{\frac{3}{2}} + 3$

Now, differentiate each term with respect to $x$:

1. Differentiate each term:

   • For the first term, $x^{-3}$:
     $\frac{d}{dx} \left[ x^{-3} \right] = -3x^{-4}$
   • For the second term, $2x^{\frac{3}{2}}$:
     $\frac{d}{dx} \left[ 2x^{\frac{3}{2}} \right] = 2 \times \frac{3}{2} x^{\frac{1}{2}} = 3x^{\frac{1}{2}}$
   • For the third term, the constant 3:
     $\frac{d}{dx} \left[ 3 \right] = 0$

2. Combine the derivatives: $\frac{dy}{dx} = -3x^{-4} + 3x^{\frac{1}{2}}$

3. Rewrite in fractional form (optional): $\frac{dy}{dx} = -\frac{3}{x^4} + \frac{3}{x^{\frac{1}{2}}}$

4. Final Simplified Form: $\frac{dy}{dx} = -3 \left[ \frac{1}{x^4} + \frac{1}{x^{\frac{5}{2}}} \right]$

Final Answer:
$\frac{dy}{dx} = -3 \left[ \frac{1}{x^4} + \frac{1}{x^{\frac{5}{2}}} \right]$

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Key Formulas or Methods Used

• Power Rule of Differentiation:
  For any term $x^n$, the derivative is:
  $\frac{d}{dx} \left[ x^n \right] = n \cdot x^{n-1}$

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Summary of Steps

1. Identify the function: $Y = x^{-3} + 2x^{\frac{3}{2}} + 3$

2. Apply the power rule to differentiate each term:

   • $\frac{d}{dx} \left[ x^{-3} \right] = -3x^{-4}$
   • $\frac{d}{dx} \left[ 2x^{\frac{3}{2}} \right] = 3x^{\frac{1}{2}}$
   • $\frac{d}{dx} \left[ 3 \right] = 0$

3. Combine the derivatives to get:
   $\frac{dy}{dx} = -3x^{-4} + 3x^{\frac{1}{2}}$

4. Rewrite in fractional form (optional):
   $\frac{dy}{dx} = -\frac{3}{x^4} + \frac{3}{x^{\frac{1}{2}}}$

5. Final simplified form:
   $\frac{dy}{dx} = -3 \left[ \frac{1}{x^4} + \frac{1}{x^{\frac{5}{2}}} \right]$