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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

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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?

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