Finding the derivative of a function is a fundamental concept in calculus. The question asks us to find the derivation of the function \(f(x) = \frac{1}{x^2}\). This involves applying the rules of differentiation, specifically the power rule, to determine the rate of change of the function.

Derivative of a Function: Understanding the Power Rule

To find the derivation of \(f(x) = \frac{1}{x^2}\), we first need to rewrite the function in a form that is easier to differentiate using the standard power rule. The power rule states that if \(f(x) = x^n\), then its derivative, denoted as \(f'(x)\) or \(\frac{d}{dx}f(x)\), is given by \(f'(x) = nx^{n-1}\).

• Step 1: Rewrite the function in power form.

  The given function is \(f(x) = \frac{1}{x^2}\). Using the property of exponents that \(\frac{1}{a^n} = a^{-n}\), we can rewrite \(f(x)\) as:

  \(f(x) = x^{-2}\)

• Step 2: Apply the power rule for differentiation.

  Now, we apply the power rule \(f'(x) = nx^{n-1}\) to \(f(x) = x^{-2}\). Here, \(n = -2\).

  \(f'(x) = (-2)x^{(-2)-1}\)

  \(f'(x) = -2x^{-3}\)

• Step 3: Rewrite the result in positive exponent form.

  Finally, to express the derivation in a more conventional format, we convert the negative exponent back to a positive one using the same exponent property \(\frac{1}{a^n} = a^{-n}\):

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

Detailed Analysis of the Derivation

Let's summarize the derivation process for \(f(x) = \frac{1}{x^2}\):

1. Start with the function: \(f(x) = \frac{1}{x^2}\)
2. Convert to exponential form: \(f(x) = x^{-2}\)
3. Apply the power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\): \(\frac{d}{dx}(x^{-2}) = (-2)x^{-2-1}\)
4. Simplify the exponent: \(= -2x^{-3}\)
5. Convert back to fractional form: \(= -\frac{2}{x^3}\)

Therefore, the derivation of \(f(x) = \frac{1}{x^2}\) is \(-\frac{2}{x^3}\).

Comparing with Options

Let's look at the given options for the derivation:

• Option 1: \(-2/x^3\) - This matches our calculated derivation.
• Option 2: \(2/x^3\) - This is incorrect as the sign is positive. The power rule correctly yields a negative sign due to \(n = -2\).
• Option 3: \(-1/2x\) - This is incorrect. It seems to imply an incorrect application of rules, possibly confusing differentiation with integration or applying a constant rule incorrectly.
• Option 4: \(1/2x\) - This is incorrect for the same reasons as Option 3, plus the sign error.

Based on our step-by-step derivation using the power rule, the correct answer is \(-\frac{2}{x^3}\).