Differentiate log(cosh(x)) + (1/2)cosh²(x) with respect to x.
Understand the Problem
The question is asking us to find the derivative of the function log(cosh(x)) + (1/2)cosh²(x) with respect to x. To solve this, we will apply the rules of differentiation including the chain rule and product rule where necessary.
Answer
The derivative is $$ \tanh(x) + \cosh(x) \sinh(x) $$
Answer for screen readers
The derivative of the function is
$$ \tanh(x) + \cosh(x) \sinh(x) $$
Steps to Solve
- Differentiate the First Term
We need to find the derivative of the function $f(x) = \log(\cosh(x))$. Using the chain rule, we have:
$$ f'(x) = \frac{1}{\cosh(x)} \cdot \frac{d}{dx}(\cosh(x)) $$
The derivative of $\cosh(x)$ is $\sinh(x)$, so we get:
$$ f'(x) = \frac{\sinh(x)}{\cosh(x)} = \tanh(x) $$
- Differentiate the Second Term
Next, we differentiate the second term $(\frac{1}{2}\cosh^2(x))$. Using the chain rule and the power rule:
$$ g(x) = \frac{1}{2} \cosh^2(x) $$ $$ g'(x) = \frac{1}{2} \cdot 2\cosh(x) \cdot \sinh(x) = \cosh(x)\sinh(x) $$
- Combine the Derivatives
Now we combine the derivatives of the two terms:
$$ \frac{d}{dx} \left( \log(\cosh(x)) + \frac{1}{2}\cosh^2(x) \right) = \tanh(x) + \cosh(x)\sinh(x) $$
Thus, the final derivative is:
$$ \tanh(x) + \cosh(x)\sinh(x) $$
The derivative of the function is
$$ \tanh(x) + \cosh(x) \sinh(x) $$
More Information
The derivative indicates the rate of change of the function $f(x)$ with respect to $x$. The functions $\sinh(x)$ and $\cosh(x)$ represent the hyperbolic sine and cosine functions, which are often used in physics and engineering.
Tips
- Forgetting to apply chain rule correctly: When differentiating $\log(\cosh(x))$, ensure to apply the chain rule properly.
- Incorrectly differentiating hyperbolic functions: Remember that the derivatives of hyperbolic functions are different from regular trigonometric functions.
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