Differentiate log(tanh(2x)) with respect to x.
Understand the Problem
The question is asking to find the derivative of the function log(tanh(2x)) with respect to the variable x. To solve this, we can use the chain rule and the properties of logarithmic and hyperbolic functions.
Answer
The derivative is $$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)}. $$
Answer for screen readers
The derivative of the function $y = \log(\tanh(2x))$ with respect to $x$ is
$$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)}. $$
Steps to Solve
- Identify the function to differentiate
We have the function $y = \log(\tanh(2x))$. To find its derivative, we will apply the chain rule, which allows us to differentiate composite functions.
- Differentiate using the chain rule
The chain rule states that if we have a function $y = \log(u)$ where $u = \tanh(2x)$, then the derivative is given by
$$ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}. $$
Thus, we have:
$$ \frac{dy}{dx} = \frac{1}{\tanh(2x)} \cdot \frac{d}{dx}(\tanh(2x)). $$
- Find the derivative of $\tanh(2x)$
We will now find $\frac{d}{dx}(\tanh(2x))$. Using the chain rule again:
$$ \frac{d}{dx}(\tanh(2x)) = \text{sech}^2(2x) \cdot \frac{d}{dx}(2x) = 2 \cdot \text{sech}^2(2x). $$
- Combine the derivatives
Now substituting back, we get:
$$ \frac{dy}{dx} = \frac{1}{\tanh(2x)} \cdot (2 \cdot \text{sech}^2(2x)). $$
Thus, we simplify this expression:
$$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)}. $$
- Final result
The final expression gives us the derivative of the original function.
The derivative of the function $y = \log(\tanh(2x))$ with respect to $x$ is
$$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)}. $$
More Information
This result employs the properties of logarithmic and hyperbolic functions. The derivative of $\tanh(x)$ is $\text{sech}^2(x)$, which is a critical concept in calculus, especially when dealing with hyperbolic functions.
Tips
- A common mistake is forgetting to apply the chain rule correctly when differentiating composite functions. To avoid this, carefully identify the outer and inner functions.
- Another mistake could be confusing $\tanh(x)$ with $\text{sech}(x)$ when differentiating; remember to clearly use the definition and derivatives of hyperbolic functions.
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