How this calculator works
Hyperbolic tangent divides hyperbolic sine by hyperbolic cosine. Its graph passes through zero, increases everywhere and approaches horizontal limits at -1 and 1. The derivative readout measures how sensitive the output is to a small change in x. The table shows this slope shrinking away from zero. All six companion values are evaluated at the same argument; reciprocal functions that divide by zero are marked undefined. Forward inputs are limited to -700 through 700 to keep sinh and cosh finite.
Formula
tanh x = sinh x / cosh x = (exp(2x)-1)/(exp(2x)+1); d(tanh x)/dx = sech²x; atanh y = ln((1+y)/(1-y))/2.
Worked example
At x = ln 2, sinh x = 3/4 and cosh x = 5/4. Therefore tanh x = 3/5 = 0.6 and the derivative is (4/5)² = 0.64. Inverse mode maps -0.6 to -ln 2.
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Frequently asked questions
Should I enter degrees or radians?
Neither conversion is applied. The argument is a dimensionless parameter rather than a circular angle.
Can tanh equal one?
For finite real x it is strictly below one. Floating-point arithmetic may round the value to one for large positive arguments; inverse mode still rejects one.
Why are coth and csch undefined at zero?
They divide by tanh(0) and sinh(0), respectively. Both denominators are zero.