Question

differentiate:
s(x) = 2x - \frac{128,000}{x^2}

Explanation:

Step1: Identify Terms to Differentiate

We have \( S'(x) = 2x - \frac{128000}{x^2} \). Rewrite \( \frac{128000}{x^2} \) as \( 128000x^{-2} \) for easier differentiation.

Step2: Apply Power Rule

The power rule is \( \frac{d}{dx}(x^n)=nx^{n - 1} \). For \( 2x \): \( n = 1 \), so derivative is \( 2\times1\times x^{0}=2 \). For \( - 128000x^{-2} \): \( n=-2 \), derivative is \( - 128000\times(-2)x^{-3}=\frac{256000}{x^{3}} \).

Step3: Combine Results

The derivative of \( S'(x) \) (let's call it \( S''(x) \)) is the sum of the derivatives of each term. So \( S''(x)=2+\frac{256000}{x^{3}} \).

Answer:

\( S''(x)=2 + \frac{256000}{x^{3}} \)