derivative of ln x ln x
Understand the Problem
The question is asking for the derivative of the function ln(x) * ln(x), which involves applying the product rule of differentiation.
Answer
\( \frac{2 \text{ln}(x)}{x} \)
Answer for screen readers
The final answer is ( \frac{2 \text{ln}(x)}{x} )
Steps to Solve
- Identify the function and apply the product rule
The given function is $ f(x) = ext{ln}(x) imes ext{ln}(x) $. Let's set $ u = ext{ln}(x) $ and $ v = ext{ln}(x) $. The product rule states that $ (uv)' = u'v + uv' $.
- Differentiate each part of the product separately
First, find the derivative of $ u = ext{ln}(x) $:
$$ u' = \frac{d}{dx} [ ext{ln}(x)] = \frac{1}{x} $$
Similarly, find the derivative of $ v = ext{ln}(x) $:
$$ v' = \frac{d}{dx} [ ext{ln}(x)] = \frac{1}{x} $$
- Apply the product rule
Apply the product rule formula $ (uv)' = u'v + uv' $ using the derivatives found in the previous step:
$$ f'(x) = u'v + uv' = \left( \frac{1}{x} \right) ext{ln}(x) + ext{ln}(x) \left( \frac{1}{x} \right) $$
Simplify the expression:
$$ f'(x) = \frac{ \text{ln}(x)}{x} + \frac{ \text{ln}(x)}{x} = 2 \cdot \frac{ \text{ln}(x)}{x} = \frac{2 \text{ln}(x)}{x} $$
The final answer is ( \frac{2 \text{ln}(x)}{x} )
More Information
The result shows that the rate of change of the function $\text{ln}(x) \cdot \text{ln}(x)$ is proportional to the natural logarithm of $x$ times 2, divided by $x$.
Tips
A common mistake is forgetting to apply the product rule when differentiating the product of two functions.
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