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Find the derivative of y = ln 2x...

Find the derivative of `y = ln 2x`

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To find the derivative of the function \( y = \ln(2x) \), we can follow these steps: ### Step 1: Rewrite the function using logarithmic properties Using the property of logarithms that states \( \ln(ab) = \ln(a) + \ln(b) \), we can rewrite the function: \[ y = \ln(2x) = \ln(2) + \ln(x) \] ### Step 2: Differentiate the function Now we differentiate both sides with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx}(\ln(2)) + \frac{d}{dx}(\ln(x)) \] Since \( \ln(2) \) is a constant, its derivative is 0. The derivative of \( \ln(x) \) is \( \frac{1}{x} \): \[ \frac{dy}{dx} = 0 + \frac{1}{x} = \frac{1}{x} \] ### Step 3: Write the final answer Thus, the derivative of \( y = \ln(2x) \) is: \[ \frac{dy}{dx} = \frac{1}{x} \]
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