d/dx ln(2x)
Understand the Problem
The question is asking for the derivative of the function ln(2x) with respect to x. To solve this, we will apply the chain rule and properties of logarithmic differentiation.
Answer
The derivative is $f'(x) = \frac{1}{x}$.
Answer for screen readers
The derivative of the function $\ln(2x)$ with respect to $x$ is $f'(x) = \frac{1}{x}$.
Steps to Solve
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Identify the function The function we need to differentiate is $f(x) = \ln(2x)$.
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Apply the chain rule To differentiate $f(x)$, we will use the chain rule. The derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$, where $u = 2x$ in this case.
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Differentiate the inner function Now we need to find the derivative of $u = 2x$. The derivative is: $$ \frac{du}{dx} = 2 $$
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Combine the derivatives Using the chain rule, we combine the derivatives: $$ f'(x) = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{2x} \cdot 2 $$
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Simplify the expression Now we simplify the expression to find the final derivative: $$ f'(x) = \frac{2}{2x} = \frac{1}{x} $$
The derivative of the function $\ln(2x)$ with respect to $x$ is $f'(x) = \frac{1}{x}$.
More Information
The derivative represents the rate of change of the function $\ln(2x)$ with respect to $x$. This particular result can be helpful in various applications in calculus, particularly in optimization problems and understanding logarithmic functions' behavior.
Tips
- Forgetting to apply the chain rule properly. Always ensure you differentiate the inner function when dealing with compositions.
- Not simplifying the final answer, which can lead to a more complicated form than necessary.
