Differentiate log(cosh(x)) + (1/2)cosh²(x) with respect to x.
Understand the Problem
The question is asking us to differentiate the function log(cosh(x)) + (1/2)cosh²(x) with respect to x. This involves applying rules of differentiation, such as the chain rule for the logarithm and the standard derivative formulas for hyperbolic functions.
Answer
$$ f'(x) = \tanh(x) + \frac{1}{2} \sinh(2x) $$
Answer for screen readers
The derivative of the function is
$$ f'(x) = \tanh(x) + \frac{1}{2} \sinh(2x) $$
Steps to Solve
- Identify the Function to Differentiate
We need to differentiate the function $f(x) = \log(\cosh(x)) + \frac{1}{2}\cosh^2(x)$ with respect to $x$.
- Differentiate the Logarithmic Term
For the first term, we apply the chain rule. The derivative of $\log(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$, where $u = \cosh(x)$.
The derivative of $\cosh(x)$ is $\sinh(x)$.
Thus, the derivative of the first term is:
$$ \frac{d}{dx} \log(\cosh(x)) = \frac{1}{\cosh(x)} \cdot \sinh(x) = \tanh(x) $$
- Differentiate the Hyperbolic Cosine Squared Term
For the second term $\frac{1}{2} \cosh^2(x)$, we can apply the chain rule. The derivative is:
$$ \frac{d}{dx} \left( \frac{1}{2} \cosh^2(x) \right) = \frac{1}{2} \cdot 2 \cosh(x) \cdot \sinh(x) = \cosh(x) \sinh(x) $$
- Combine the Results
Now, we can combine the derivatives from both terms:
$$ f'(x) = \tanh(x) + \cosh(x) \sinh(x) $$
- Simplify if Necessary
We can express the term $\cosh(x) \sinh(x)$ in terms of $\sinh(2x)$ using the identity $\sinh(2x) = 2 \sinh(x) \cosh(x)$:
$$ f'(x) = \tanh(x) + \frac{1}{2} \sinh(2x) $$
The derivative of the function is
$$ f'(x) = \tanh(x) + \frac{1}{2} \sinh(2x) $$
More Information
The derivative $f'(x)$ gives the rate of change of the function with respect to $x$. This information is important in various areas such as physics and engineering, where understanding how functions change is crucial for modeling real-world phenomena.
Tips
- Forgetting to apply the chain rule when differentiating composite functions, such as $\log(\cosh(x))$.
- Ignoring the derivatives of hyperbolic functions, which can often be confused with their trigonometric counterparts.
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