Question

If $ f(x) = \frac{1}{x^2} $, $ u = f(x) $, and $ f'(x) $, then find $ \frac{du}{dx} $.

• A. \( -\frac{2}{x^3} \)
• B. \( \frac{2}{x^3} \)
• C. \( -\frac{1}{x^3} \)
• D. \( \frac{1}{x^3} \)

Hint:

For functions of the form \( \frac{1}{x^n} \), the derivative is \( -n x^{-(n+1)} \). Keep this in mind when differentiating rational powers of \( x \).

Answer 1

The Correct Option is B

We are given: \[ f(x) = \frac{1}{x^2} \] Now, we need to differentiate \( f(x) \) to find \( f'(x) \): \[ f'(x) = \frac{d}{dx} \left( \frac{1}{x^2} \right) \] Using the power rule for differentiation \( \frac{d}{dx} x^n = n x^{n-1} \), we get: \[ f'(x) = -2x^{-3} = -\frac{2}{x^3} \]
Thus, \( \frac{du}{dx} = f'(x) = -\frac{2}{x^3} \).