Differentiate cosh²(5x) with respect to x.

Steps to Solve

1. Identify the function to differentiate

We start with the function ( f(x) = \cosh^2(5x) ). This is a composition of functions, which means we will apply the chain rule.

2. Apply the chain rule

The chain rule states that if we have a function ( g(h(x)) ), then the derivative is given by:

$$ g'(h(x)) \cdot h'(x) $$

Here, let ( g(u) = u^2 ) where ( u = \cosh(5x) ). Therefore, we have:

$$ \frac{d}{dx} \cosh^2(5x) = 2 \cosh(5x) \cdot \frac{d}{dx} (\cosh(5x)) $$

3. Differentiate the inner function

Next, we differentiate ( \cosh(5x) ). The derivative of the hyperbolic cosine function is:

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

Using the chain rule again, we have:

$$ \frac{d}{dx} \cosh(5x) = 5 \sinh(5x) $$

4. Combine the derivatives

Now, we can combine our results:

$$ \frac{d}{dx} \cosh^2(5x) = 2 \cosh(5x) \cdot (5 \sinh(5x)) $$

So we simplify this to:

$$ \frac{d}{dx} \cosh^2(5x) = 10 \cosh(5x) \sinh(5x) $$