Proof
The hyperbolic cosine function is defined as:
To find the derivative, we differentiate using the sum rule:
Differentiating each term:
This simplifies to:
Thus, the derivative of is:
Explanation
The hyperbolic cosine function, , is defined as . This formula combines the exponential functions and .
To differentiate , we use basic differentiation rules. The function can be broken down into . Here, 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 is simply , and the derivative of is due to the chain rule.
Putting these results together, the derivative becomes . This expression is the definition of , the hyperbolic sine function. Therefore, the derivative of cosh(x) is sinh(x).
Q.E.D.