We want to verify if the derivative of $\coth(x)$ is equal to $-\csch^2(x)$.

The hyperbolic cotangent function is defined as:

$\coth(x) = \frac{\cosh(x)}{\sinh(x)}$

To differentiate $\coth(x)$, we use the quotient rule, which is given by:

$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

For $\coth(x) = \frac{\cosh(x)}{\sinh(x)}$, let:

• $f(x) = \cosh(x)$
• $g(x) = \sinh(x)$

We know that:

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

Now applying the quotient rule:

$\frac{d}{dx} \left( \coth(x) \right) = \frac{\sinh(x) \cdot \sinh(x) - \cosh(x) \cdot \cosh(x)}{[\sinh(x)]^2}$

This simplifies to:

$\frac{d}{dx} \left( \coth(x) \right) = \frac{\sinh^2(x) - \cosh^2(x)}{[\sinh(x)]^2}$

Using the hyperbolic identity $\cosh^2(x) - \sinh^2(x) = 1$, this becomes:

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

Since $\csch(x) = \frac{1}{\sinh(x)}$, we can rewrite this as:

$\frac{d}{dx} \left( \coth(x) \right) = -\csch^2(x)$

Thus, we have verified that:

$\frac{d}{dx} \left( \coth(x) \right) = -\csch^2(x)$

Do you want more details or have any questions?

Here are 5 related questions:

1. How do you differentiate other hyperbolic functions like $\cosh(x)$ and $\sinh(x)$?
2. What is the relationship between hyperbolic and trigonometric functions?
3. Can you show how the chain rule applies to hyperbolic functions?
4. How do you integrate $\csch^2(x)$?
5. What are the applications of hyperbolic functions in calculus?

Tip: Remember that hyperbolic identities