Differentiate log | sinh x | with respect to x
Understand the Problem
The question is asking for the differentiation of the logarithm of the absolute value of the hyperbolic sine function with respect to the variable x. This involves applying the chain rule and knowing how to differentiate logarithmic and hyperbolic functions.
Answer
$$ \frac{d}{dx}(\ln|\sinh(x)|) = \coth(x) $$
Answer for screen readers
The derivative of the logarithm of the absolute value of the hyperbolic sine function with respect to $x$ is:
$$ \frac{d}{dx}(\ln|\sinh(x)|) = \coth(x) $$
Steps to Solve
- Differentiate the outer function We start by using the chain rule to differentiate the logarithm function. The derivative of $\ln|u|$ is given by:
$$ \frac{d}{dx}(\ln|u|) = \frac{1}{u} \cdot \frac{du}{dx} $$
In this case, $u = \sinh(x)$, so the derivative is:
$$ \frac{d}{dx}(\ln|\sinh(x)|) = \frac{1}{\sinh(x)} \cdot \frac{d}{dx}(\sinh(x)) $$
- Differentiate the hyperbolic sine function Next, we need to differentiate $\sinh(x)$ with respect to $x$. The derivative is:
$$ \frac{d}{dx}(\sinh(x)) = \cosh(x) $$
- Combine the results Now we can substitute this result back into the derivative from step 1:
$$ \frac{d}{dx}(\ln|\sinh(x)|) = \frac{1}{\sinh(x)} \cdot \cosh(x) $$
This simplifies to:
$$ \frac{\cosh(x)}{\sinh(x)} $$
- Simplify the expression using hyperbolic identities Recall that $\frac{\cosh(x)}{\sinh(x)}$ is the definition of the hyperbolic cotangent:
$$ \frac{\cosh(x)}{\sinh(x)} = \coth(x) $$
Therefore, the final answer is:
$$ \frac{d}{dx}(\ln|\sinh(x)|) = \coth(x) $$
The derivative of the logarithm of the absolute value of the hyperbolic sine function with respect to $x$ is:
$$ \frac{d}{dx}(\ln|\sinh(x)|) = \coth(x) $$
More Information
This differentiation showcases the interplay between logarithmic and hyperbolic functions. The hyperbolic cotangent, $\coth(x)$, is a fundamental function in various fields such as calculus, physics, and engineering, particularly in contexts related to hyperbolic geometries and population dynamics.
Tips
- Forgetting to apply the chain rule correctly when differentiating composite functions.
- Overlooking the absolute value in the logarithmic differentiation, which can lead to not considering negative values of $\sinh(x)$.
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