Derivative of sinh x
How to find the derivative of the given function by using the derivative of sinh x: formula, 1 example, and its solution.
The hyperbolic functions are functions
that look like trigonometric functions
with the letter h:
sinh x, cosh x, tanh x, ... .
sinh x = (ex - e-x)/2
It's read as [hyperbolic sine] or [sin-ch].
cosh x = (ex + e-x)/2
It's read as [hyperbolic cosine] or [co-sh].
There's a reason
these are called hyperbolic functions.
a point (cos θ, sin θ)
is always on the unit circle x2 + y2 = 1.
Similar to this,
a point ((ex + e-x)/2, (ex - e-x)/2)
is always on the unit hyperbola x2 - y2 = 1.
So (cosh x, sinh x) is defined as ((ex + e-x)/2, (ex - e-x)/2).
This is why sinh x = (ex - e-x)/2 and cosh x = (ex + e-x)/2.
The derivative of sinh x, [sinh x]',
is cosh x.