sinh(x)=(expo(x)-expo(-x))/2 and cosh(x)= (expo(x)+expo(-x))/2. find the first derivatives and explain

The hyperbolic sine and cosine functions are defined as follows:

sinh(x) = (e^x - e^-x) / 2 cosh(x) = (e^x + e^-x) / 2

To find the first derivative of these functions, we use the chain rule and the fact that the derivative of e^x is e^x and the derivative of e^-x is -e^-x.

For sinh(x), the derivative is:

d/dx sinh(x) = d/dx [(e^x - e^-x) / 2] = (1/2) * (d/dx e^x - d/dx e^-x) = (1/2) * (e^x - (-e^-x)) = (1/2) * (e^x + e^-x) = cosh(x)

So, the derivative of sinh(x) is cosh(x).

For cosh(x), the derivative is:

d/dx cosh(x) = d/dx [(e^x + e^-x) / 2] = (1/2) * (d/dx e^x + d/dx e^-x) = (1/2) * (e^x - (-e^-x)) = (1/2) * (e^x - e^-x) = sinh(x)

So, the derivative of cosh(x) is sinh(x).

In summary, the derivative of sinh(x) is cosh(x) and the derivative of cosh(x) is sinh(x). This is similar to the relationship between the derivatives of the standard sine and cosine functions, where the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).