Derivative of sinh(x) - Proof and Explanation

Proof

The hyperbolic sine function is defined as:

sinh(x)=exex2

To find the derivative, we use the derivative of exponential functions. The derivative of ex is ex, and the derivative of ex is ex.

Now, differentiating sinh(x):

ddx(exex2)=12(ex(ex))

This simplifies to:

12(ex+ex)=cosh(x)

Thus, the derivative of sinh(x) is:

ddxsinh(x)=cosh(x)

Explanation

The hyperbolic sine function, sinh(x), is similar to the sine function but based on exponential functions. It is defined as:

sinh(x)=exex2

This expression represents the difference between the exponential growth ex and the exponential decay ex, divided by two.

To find the derivative, we differentiate each part of the function. The derivative of ex with respect to x is ex, and the derivative of ex with respect to x is ex. This is because of the chain rule, where the derivative of x is 1.

Substituting these derivatives into the formula for sinh(x):

ddx(exex2)=12(ex(ex))

This simplifies to:

12(ex+ex)

This expression is exactly the definition of cosh(x), the hyperbolic cosine function:

ex+ex2=cosh(x)

Q.E.D.