Does anyone know how I get from
y=2 tan-1 (et)
to
dy/dx = sech t ?
(the minus one after tan and the t following the e are powers!!)
solve for e^t (not really necessary, but I figure you'll see it better this way)
take the derivative
solve for y'
substitute for y from the first equation
simplify (draw a triangle if necessary... or just memorize the derivative of arctangent and save yourself this extra work
divide numerator and denominator by e^t
an equation is equal to 1 over it's reciprocal
the denominator is the equation for cosh(t)
and simplify
sech(x) = 1 / cosh(x)
We know that the derivative of cosh(x) is sinh(x).
The derivative of something like 1/f(x) is -f'(x)/f²(x)
---> the derivative of sech(x) is
You can take a look at these explanations, here : Hyperbolic Secant -- from Wolfram MathWorld