Differentiate √(cosh²(x) - 1) with respect to x.

Steps to Solve

1. Identify the function to differentiate

We need to find the derivative of the function ( y = \sqrt{\cosh^2(x) - 1} ).

2. Apply the chain rule

Using the chain rule, we differentiate the outer function ( \sqrt{u} ) where ( u = \cosh^2(x) - 1 ). The derivative of ( \sqrt{u} ) is:

$$ \frac{dy}{du} = \frac{1}{2\sqrt{u}} $$

So we have:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{2\sqrt{\cosh^2(x) - 1}} \cdot \frac{du}{dx} $$

3. Differentiate the inner function

Next, we need to differentiate ( u = \cosh^2(x) - 1 ).

Using the chain rule again, with ( v = \cosh(x) ):

$$ \frac{du}{dx} = 2\cosh(x) \cdot \sinh(x) $$

4. Combine the results

Now, substitute ( \frac{du}{dx} ) into our ( \frac{dy}{dx} ) expression:

$$ \frac{dy}{dx} = \frac{1}{2\sqrt{\cosh^2(x) - 1}} \cdot 2\cosh(x) \cdot \sinh(x) $$

This simplifies to:

$$ \frac{dy}{dx} = \frac{\cosh(x) \cdot \sinh(x)}{\sqrt{\cosh^2(x) - 1}} $$

5. Final simplification

The expression can be simplified further by recognizing the identity ( \cosh^2(x) - 1 = \sinh^2(x) ):

Thus,

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

This simplifies to:

$$ \frac{dy}{dx} = \cosh(x) $$