Question

Let \(f(x) = \frac{1}{x^2}\) and let \(u = f(x)f''(x)\). Then \(\frac{du}{dx} =\)

• A. \(-36x^{-7}\)
• B. \(36x^{-7}\)
• C. \(42x^{-7}\)
• D. \(-42x^{-7}\)
• E. \(-30x^{-7}\)

Hint:

Remember power rule: \(\frac{d}{dx} x^n = n x^{n-1}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Find \(f'(x)\) and \(f''(x)\), then compute \(u = f(x)f''(x)\), then differentiate \(u\).

Step 2: Detailed Explanation:
\(f(x) = x^{-2}\)
\(f'(x) = -2x^{-3}\)
\(f''(x) = 6x^{-4}\)
\(u = f(x)f''(x) = (x^{-2})(6x^{-4}) = 6x^{-6}\)
\(\frac{du}{dx} = 6 \cdot (-6)x^{-7} = -36x^{-7}\)

Step 3: Final Answer:
\(\frac{du}{dx} = -36x^{-7}\).