What is the derivative of 2 ln x?
Understand the Problem
The question is asking for the derivative of the function 2 ln(x), which involves applying differentiation rules to find the rate of change of the function with respect to x.
Answer
\( \frac{2}{x} \)
Answer for screen readers
The final answer is ( \frac{2}{x} )
Steps to Solve
- Identify the function to differentiate
The given function is $f(x) = 2 \ln(x)$.
- Apply the constant multiple rule
The constant multiple rule in differentiation states that the derivative of a constant times a function is the constant times the derivative of the function: [ \frac{d}{dx} [c \cdot f(x)] = c \cdot \frac{d}{dx} [f(x)] ] Here, $c = 2$ and $f(x) = \ln(x)$.
- Differentiate the natural logarithm function
Recall that the derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$: [ \frac{d}{dx} [\ln(x)] = \frac{1}{x} ]
- Combine the results
Multiply the constant $2$ by the derivative of $\ln(x)$: [ \frac{d}{dx} [2 \ln(x)] = 2 \cdot \frac{1}{x} = \frac{2}{x} ]
The final answer is ( \frac{2}{x} )
More Information
Differentiation of logarithmic functions is a common operation in calculus, especially in problems involving growth rates and elasticity in economics.
Tips
A common mistake is to neglect the constant multiple rule when differentiating, leading to an incorrect answer. Always remember to apply the rule before differentiating the function.
