Differentiate log(cosh(x)) + (1/2)cosh²(x) with respect to x.

Steps to Solve

1. Differentiate the first term

To differentiate the first term, $\log(\cosh(x))$, we will use the chain rule. The derivative of $\log(u)$ is $\frac{1}{u} \frac{du}{dx}$. Here, $u = \cosh(x)$.

$$ \frac{d}{dx}(\log(\cosh(x))) = \frac{1}{\cosh(x)} \cdot \frac{d}{dx}(\cosh(x)) $$

The derivative of $\cosh(x)$ is $\sinh(x)$, so we have:

$$ \frac{d}{dx}(\log(\cosh(x))) = \frac{\sinh(x)}{\cosh(x)} = \tanh(x) $$

2. Differentiate the second term

Next, we differentiate the second term, which is $\frac{1}{2} \cosh^2(x)$. We can use the chain rule and the power rule here. The derivative of $u^n$ where $u = \cosh(x)$ and $n=2$ gives $n u^{n-1} \cdot \frac{du}{dx}$.

$$ \frac{d}{dx}\left(\frac{1}{2} \cosh^2(x)\right) = \frac{1}{2} \cdot 2 \cosh(x) \cdot \frac{d}{dx}(\cosh(x)) $$

Again, since $\frac{d}{dx}(\cosh(x)) = \sinh(x)$, we have:

$$ \frac{d}{dx}\left(\frac{1}{2} \cosh^2(x)\right) = \cosh(x) \sinh(x) $$

3. Combine the derivatives

Now, we combine the results from both derivatives to find the overall derivative of the expression:

$$ \frac{d}{dx}\left(\log(\cosh(x)) + \frac{1}{2} \cosh^2(x)\right) = \tanh(x) + \cosh(x) \sinh(x) $$

4. Final Formulation

The final simplified derivative of the entire expression is:

$$ \frac{d}{dx}\left(\log(\cosh(x)) + \frac{1}{2} \cosh^2(x)\right) = \tanh(x) + \frac{1}{2} \sinh(2x) $$