Proof
The hyperbolic sine function is defined as:
To find the derivative, we use the derivative of exponential functions. The derivative of is , and the derivative of is .
Now, differentiating :
This simplifies to:
Thus, the derivative of is:
Explanation
The hyperbolic sine function, , is similar to the sine function but based on exponential functions. It is defined as:
This expression represents the difference between the exponential growth and the exponential decay , divided by two.
To find the derivative, we differentiate each part of the function. The derivative of with respect to is , and the derivative of with respect to is . This is because of the chain rule, where the derivative of is .
Substituting these derivatives into the formula for :
This simplifies to:
This expression is exactly the definition of , the hyperbolic cosine function:
Q.E.D.