Derivative of cosh(x) - Proof and Explanation

Proof

The hyperbolic cosine function is defined as:

cosh(x)=ex+ex2

To find the derivative, we differentiate using the sum rule:

ddxcosh(x)=ddx(ex+ex2)

Differentiating each term:

12(ddxex+ddxex) =12(exex)

This simplifies to:

sinh(x)

Thus, the derivative of cosh(x) is:

ddxcosh(x)=sinh(x)

Explanation

The hyperbolic cosine function, cosh(x), is defined as ex+ex2. This formula combines the exponential functions ex and ex.

To differentiate cosh(x), we use basic differentiation rules. The function can be broken down into 12(ex+ex). Here, 12 is a constant factor that we can factor out during differentiation.

We then apply the differentiation rule to each part of the expression. The derivative of ex is simply ex, and the derivative of ex is ex due to the chain rule.

Putting these results together, the derivative becomes 12(exex). This expression is the definition of sinh(x), the hyperbolic sine function. Therefore, the derivative of cosh(x) is sinh(x).

Q.E.D.