I am trying to find the equation of tangent line of the curve that pass through the origin. The equation of the curve is y = tanh(×). I am solving this in hopes of solving the critical value of positive constant c for which cx = tanh(x) has nontrivial solutions
2026-03-25 01:17:17.1774401437
Finding tangent line of curve that pass through the origin
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Derivative of $\tanh(x)$ simply is $sech^{2}x$, with slope$=1$ at origin. The tangent has equation.
$$ y= x $$
For $c<1$ the second equation gives two roots as anti-symmetric solutions found by direct substitution.
If you are trying to find points of intersection in graphs of
$$ y=cx, \quad y= \tanh x $$
then there is no need at all to worry about slope at origin.
Did you try to solve numerically using Newton-Raphson iteration?