Questions: Find the derivative of f(x) = 2 / (3x). f'(x) = -2 / (3x^2)

Solution Steps

Given the function \( f(x) = \frac{2}{3x} \), we can rewrite it to explicitly show the negative exponent: \[ f(x) = 2 \cdot (3x)^{-1} \]

Using the power rule for differentiation, which states that \(\frac{d}{dx} [x^n] = n \cdot x^{n-1}\), we differentiate the function: \[ f'(x) = 2 \cdot \frac{d}{dx} [(3x)^{-1}] \]

First, we differentiate the inner function \((3x)^{-1}\): \[ \frac{d}{dx} [(3x)^{-1}] = -1 \cdot (3x)^{-2} \cdot 3 = -3 \cdot (3x)^{-2} \]

Now, we multiply by the constant 2: \[ f'(x) = 2 \cdot (-3) \cdot (3x)^{-2} = -2 \cdot (3x)^{-2} \]

Simplify the expression to match one of the provided options: \[ f'(x) = \frac{-2}{3x^2} \]

Final Answer