Derivative of asinh(x) - Proof and Explanation

Proof

We start by defining y=asinh(x). By definition, the inverse hyperbolic sine function means:

x=sinh(y)

where sinh(y)=eyey2.

To find the derivative of asinh(x), we first implicitly differentiate x=sinh(y) with respect to x:

1=cosh(y)dydx

Here, cosh(y) is the hyperbolic cosine function, which is defined as:

cosh(y)=ey+ey2

Rearranging for dydx, we get:

dydx=1cosh(y)

We now need to express cosh(y) in terms of x. Using the identity cosh2(y)sinh2(y)=1, we get:

cosh2(y)=1+sinh2(y)

Since sinh(y)=x, this becomes:

cosh2(y)=1+x2

Thus, cosh(y)=1+x2. Substituting this back into the expression for dydx, we obtain:

dydx=11+x2

Hence, the derivative of asinh(x) is:

ddxasinh(x)=11+x2

Explanation

To understand this derivative, we first recognize that asinh(x) is the inverse hyperbolic sine function, which means if y=asinh(x), then x=sinh(y). The hyperbolic sine function, sinh(y), is defined as eyey2.

To find the derivative of asinh(x), we differentiate the equation x=sinh(y) implicitly with respect to x. Differentiating both sides, we get 1=cosh(y)dydx, where cosh(y) is the hyperbolic cosine function defined as ey+ey2.

Solving for dydx, we have dydx=1cosh(y). Next, we need to express cosh(y) in terms of x. Using the identity cosh2(y)sinh2(y)=1, we substitute sinh(y)=x, giving cosh2(y)=1+x2. This means cosh(y)=1+x2.

Substituting cosh(y)=1+x2 back into our expression for dydx, we find dydx=11+x2. Therefore, the derivative of asinh(x) with respect to x is 11+x2.

Q.E.D.