Question

\(\frac{d}{dx}(\frac{1}{x}\frac{d^2}{dx^2}(\frac{1}{x^3}))=\)

• A. -36x^-7
• B. 36x^-7
• C. 72x^-6
• D. 72x^-7
• E. -72x^-7

Answer 1

Answer: (E)

We are asked to compute: \[ \frac{d}{dx} \left( \frac{1}{x} \frac{d^2}{dx^2} \left( \frac{1}{x^3} \right) \right) \] Step 1: Compute the second derivative of \( \frac{1}{x^3} \). We start with: \[ y = \frac{1}{x^3} = x^{-3} \] The first derivative of \( y \) is: \[ \frac{dy}{dx} = -3x^{-4} \] Now, the second derivative: \[ \frac{d^2y}{dx^2} = 12x^{-5} \] Step 2: Now, compute \( \frac{1}{x} \times \frac{d^2}{dx^2} \left( \frac{1}{x^3} \right) \): \[ \frac{1}{x} \times 12x^{-5} = 12x^{-6} \] Step 3: Finally, differentiate with respect to \( x \): \[ \frac{d}{dx} \left( 12x^{-6} \right) = -72x^{-7} \]

The correct option is (E) : \(-72x^{-7}\)

Answer 2

• We want to find: \[\frac{d}{dx}\left(\frac{1}{x}\frac{d^2}{dx^2}\left(\frac{1}{x^3}\right)\right)\]
• First, rewrite \(\frac{1}{x^3}\) as \(x^{-3}\). Then find the first derivative: \[\frac{d}{dx}(x^{-3}) = -3x^{-4}\]
• Find the second derivative: \[\frac{d^2}{dx^2}(x^{-3}) = \frac{d}{dx}(-3x^{-4}) = 12x^{-5}\]
• Now, multiply by \(\frac{1}{x}\): \[\frac{1}{x}\frac{d^2}{dx^2}\left(\frac{1}{x^3}\right) = \frac{1}{x}(12x^{-5}) = 12x^{-6}\]
• Finally, differentiate with respect to \(x\): \[\frac{d}{dx}(12x^{-6}) = 12(-6)x^{-7} = -72x^{-7}\]
• Therefore, \(\frac{d}{dx}(\frac{1}{x}\frac{d^2}{dx^2}(\frac{1}{x^3})) = -72x^{-7}\).