Proof
To find the derivative of , we start with its definition:
Using the quotient rule, which states that the derivative of a quotient is:
Let and . The derivatives are and .
Applying the quotient rule:
Simplifying the numerator:
Thus, the derivative becomes:
So, the derivative of is:
Explanation
To understand the derivative of , we begin by recognizing its definition as the hyperbolic cotangent function, expressed as . This function represents the ratio of the hyperbolic cosine to the hyperbolic sine.
We use the quotient rule to find the derivative of a quotient of two functions. According to this rule, for a function , the derivative is , where and are both functions of .
In the case of , we set and . The derivative of is , and the derivative of is .
Applying these derivatives in the quotient rule, we have:
The numerator simplifies to . According to the identity , we know that .
Thus, the expression becomes:
Since is , the final result is:
Q.E.D.