Proof
We start by defining . By definition, the inverse hyperbolic sine function means:
where .
To find the derivative of , we first implicitly differentiate with respect to :
Here, is the hyperbolic cosine function, which is defined as:
Rearranging for , we get:
We now need to express in terms of . Using the identity , we get:
Since , this becomes:
Thus, . Substituting this back into the expression for , we obtain:
Hence, the derivative of is:
Explanation
To understand this derivative, we first recognize that is the inverse hyperbolic sine function, which means if , then . The hyperbolic sine function, , is defined as .
To find the derivative of , we differentiate the equation implicitly with respect to . Differentiating both sides, we get , where is the hyperbolic cosine function defined as .
Solving for , we have . Next, we need to express in terms of . Using the identity , we substitute , giving . This means .
Substituting back into our expression for , we find . Therefore, the derivative of with respect to is .
Q.E.D.