Remember the curve of sinh(x)? Here it is.
Recall that as x gets very large, sinh(x) grows like ex/2. Since the derivative of ex is just ex we expect that the derivative of sinh(x) will also grow like ex/2 as x gets large.
Now what about as x gets very negative?
Remember sinh(x) looks like -e-x/2 out there. The derivative of that is just e-x/2 (the minus sign that comes down cancelling with the one at the front). So although sinh(x) is very negative when x is, the derivative of sinh(x) has to be very positive, like e-x/2. This fits in with the picture above - the slope is positive everywhere, both for positive and negative values of x.
In summary, for the derivative of sinh(x) we're expecting a function that grows exponentially at both ends. Here it is:
Yes it's cosh(x). Again we see the similarity with the trig functions - just as the derivative of sin(x) is cos(x), so the derivative of sinh(x) is cosh(x).
We can easily check that this is indeed the case by using the definition of sinh(x) in terms of exponential functions:
Since the derivative of ex is still ex and the derivative of e-x is -e-x, we can differentiate these equations, to find the following two results:
So the derivative of sinh(x) is cosh(x) and the derivative of cosh(x) is sinh(x). Notice that this is UNLIKE the trig version, where the derivative of cos(x) is MINUS sin(x).
Here's the curve of cosh(x) again, to show you that the derivative must be sinh(x) rather than minus sinh(x).
Look at positive values of x for instance. The slope is positive, i.e. the function increases as x increases. But -sinh(x) is negative for positive values of x.
If we have sinh(2x) instead of sinh(x), we treat the 2 inside just as we do with other functions, so the derivative of sinh(2x) is 2cosh(2x).
Similarly the derivative of sinh(x2) is 2xsinh(x2) - that's a so-called nested function, or a "function of a function".
If you are not familiar with differentiating nested functions, products of functions, or quotients of functions, have a look at the MathHelp notebook on Differentiation.