If anyone can explain these, I would appreciate it. They are pretty basic but I don't get them. If anyone knows any or all, would be much appreciated.
Numerical value of: tanh(1)
cosh(ln3)
sech(0)
Proving identity of sinhx +coshx=e^x
Derivative of tanh(1 + e^2x)
sinh(coshx)
Numerical value of: tanh(1)
cosh(ln3)
sech(0)
Proving identity of sinhx +coshx=e^x
Derivative of tanh(1 + e^2x)
sinh(coshx)
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cosh(x) = (1/2) * (e^(x) + e^(-x))
sinh(x) = (1/2) * (e^(x) - e^(-x))
tanh(x) = (e^(x) - e^(-x)) / (e^(x) + e^(-x))
tanh(1) = (e^1 - e^(-1)) / (e^(1) + e^(-1))
tanh(1) = (e - 1/e) / (e + 1/e)
tanh(1) = (e^2 - 1) / (e^2 + 1)
cosh(ln(3)) =>
(1/2) * (e^(ln(3)) + e^(-ln(3))) =>
(1/2) * (3 + 1/3) =>
(1/2) * (9/3 + 1/3) =>
(1/2) * (10/3) =>
5/3
sech(0) =>
1 / cosh(0) =>
2 / (e^(x) + e^(-x)) =>
2 / (e^0 + e^0) =>
2 / (2 * 1) =>
1
sinh(x) + cosh(x) =>
(1/2) * (e^(x) - e^(-x)) + *(1/2) * (e^(x) + e^(-x)) =>
(1/2) * (e^(x) - e^(-x) + e^(x) + e^(-x)) =>
(1/2) * (2 * e^(x)) =>
e^(x)
tanh(1 + e^(2x)) =>
(e^(1 + e^(2x)) - e^(-(1 + e^(2x))) / (e^(1 + e^(2x)) + e^(-(1 + e^(2x))) =>
(e^(2 + 2 * e^(2x)) - 1) / (e^(2 + 2 * e^(2x)) + 1)
u = e^(2 + 2 * e^(2x)) - 1)
du = e^(2 + 2 * e^(2x)) * (2 * 2 * e^(2x)) = 4 * e^(2x) * e^(2 * (1 + e^(2x)))
v = e^(2 + 2 * e^(2x)) + 1
dv = 4 * e^(2x) * e^(2 * (1 + e^(2x)))
(vdu - udv) / (v^2) =>
4 * e^(2x) * e^(2 * (1 + e^(2x))) * (e^(2 + 2 * e^(2x)) + 1 - e^(2 + 2 * e^(2x)) + 1) / (e^(2 + 2 * e^(2x)) + 1)^2 =>
4 * e^(2x) * e^(2 * (1 + e^(2x))) * (2) / (e^(2 * (1 + e^(2x))) + 1)^2
8 * e^(2x) * e^(2 * (1 + e^(2x))) / (e^(2 * (1 + e^(2x))) + 1)^2
sinh(cosh(x)) =>
(1/2) * (e^(cosh(x)) - e^(-cosh(x))) =>
(1/2) * (e^((1/2) * (e^(x) + e^(-x)))) - e^((-1/2) * (e^(x) + e^(-x))))
(1/2) * (e^((1/2) * ((e^(2x) + 1) / e^x)) - 1 / e^((1/2) * ((e^(2x) + 1) / e^(x)))
sinh(x) = (1/2) * (e^(x) - e^(-x))
tanh(x) = (e^(x) - e^(-x)) / (e^(x) + e^(-x))
tanh(1) = (e^1 - e^(-1)) / (e^(1) + e^(-1))
tanh(1) = (e - 1/e) / (e + 1/e)
tanh(1) = (e^2 - 1) / (e^2 + 1)
cosh(ln(3)) =>
(1/2) * (e^(ln(3)) + e^(-ln(3))) =>
(1/2) * (3 + 1/3) =>
(1/2) * (9/3 + 1/3) =>
(1/2) * (10/3) =>
5/3
sech(0) =>
1 / cosh(0) =>
2 / (e^(x) + e^(-x)) =>
2 / (e^0 + e^0) =>
2 / (2 * 1) =>
1
sinh(x) + cosh(x) =>
(1/2) * (e^(x) - e^(-x)) + *(1/2) * (e^(x) + e^(-x)) =>
(1/2) * (e^(x) - e^(-x) + e^(x) + e^(-x)) =>
(1/2) * (2 * e^(x)) =>
e^(x)
tanh(1 + e^(2x)) =>
(e^(1 + e^(2x)) - e^(-(1 + e^(2x))) / (e^(1 + e^(2x)) + e^(-(1 + e^(2x))) =>
(e^(2 + 2 * e^(2x)) - 1) / (e^(2 + 2 * e^(2x)) + 1)
u = e^(2 + 2 * e^(2x)) - 1)
du = e^(2 + 2 * e^(2x)) * (2 * 2 * e^(2x)) = 4 * e^(2x) * e^(2 * (1 + e^(2x)))
v = e^(2 + 2 * e^(2x)) + 1
dv = 4 * e^(2x) * e^(2 * (1 + e^(2x)))
(vdu - udv) / (v^2) =>
4 * e^(2x) * e^(2 * (1 + e^(2x))) * (e^(2 + 2 * e^(2x)) + 1 - e^(2 + 2 * e^(2x)) + 1) / (e^(2 + 2 * e^(2x)) + 1)^2 =>
4 * e^(2x) * e^(2 * (1 + e^(2x))) * (2) / (e^(2 * (1 + e^(2x))) + 1)^2
8 * e^(2x) * e^(2 * (1 + e^(2x))) / (e^(2 * (1 + e^(2x))) + 1)^2
sinh(cosh(x)) =>
(1/2) * (e^(cosh(x)) - e^(-cosh(x))) =>
(1/2) * (e^((1/2) * (e^(x) + e^(-x)))) - e^((-1/2) * (e^(x) + e^(-x))))
(1/2) * (e^((1/2) * ((e^(2x) + 1) / e^x)) - 1 / e^((1/2) * ((e^(2x) + 1) / e^(x)))