Differentiate log(tanh(2x)) with respect to x.
Understand the Problem
The question is asking us to find the derivative of the function log(tanh(2x)) with respect to x. This involves using the chain rule and the derivative of the hyperbolic tangent function.
Answer
$$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)} $$
Answer for screen readers
The derivative of ( \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 need to differentiate the function ( y = \log(\tanh(2x)) ) with respect to ( x ).
- Apply the chain rule
To differentiate ( y = \log(u) ) where ( u = \tanh(2x) ), we use the chain rule: $$ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} $$
- Differentiate the inner function
Now we need to differentiate ( u = \tanh(2x) ). The derivative of the hyperbolic tangent is: $$ \frac{du}{dx} = \text{sech}^2(2x) \cdot \frac{d}{dx}(2x) = 2 \cdot \text{sech}^2(2x) $$
- Substitute back into the chain rule
Now substitute ( u ) and ( \frac{du}{dx} ) back into the chain rule: $$ \frac{dy}{dx} = \frac{1}{\tanh(2x)} \cdot 2 \cdot \text{sech}^2(2x) $$
- Simplify the expression
We can write the final result as: $$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)} $$
The derivative of ( \log(\tanh(2x)) ) with respect to ( x ) is $$ \frac{dy}{dx} = \frac{2 \cdot \text{sech}^2(2x)}{\tanh(2x)} $$
More Information
This result involves the use of both the logarithmic differentiation and the hyperbolic functions' properties. It's interesting to note that the hyperbolic functions are analogous to trigonometric functions but apply to hyperbolas instead of circles.
Tips
- Forgetting to use the chain rule correctly, particularly when differentiating composite functions.
- Confusing hyperbolic functions with their trigonometric counterparts; it's important to remember their distinct derivatives.
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