Derivative of tanh x: Formula, Proof and Examples

The derivative of \tanh x is \operatorname{sech}^2 x, which you can also write as 1 - \tanh^2 x.

Here \operatorname{sech} x = \frac{1}{\cosh x}, so \operatorname{sech}^2 x = \frac{1}{\cosh^2 x}.

Proof with the quotient rule

Write \tanh x = \frac{\sinh x}{\cosh x} and use the quotient rule, with the derivative of sinh x being \cosh x and the derivative of cosh x being \sinh x:

The hyperbolic identity \cosh^2 x - \sinh^2 x = 1 simplifies the top:

Splitting the fraction instead gives the other form: \frac{\cosh^2 x}{\cosh^2 x} - \frac{\sinh^2 x}{\cosh^2 x} = 1 - \tanh^2 x.

Graph of tanh x and its derivative

The derivative is always positive, so \tanh x always increases. But it never reaches \pm 1; those are horizontal asymptotes. Compare the derivative of arctan x, another S-curve with slope 1 at the origin, though arctan approaches its asymptotes much more slowly.

Comparing tan and tanh

The sign flip comes from the identities: \sec^2 x = 1 + \tan^2 x but \operatorname{sech}^2 x = 1 - \tanh^2 x.

Chain rule version and examples

Example 1. \frac{d}{dx}\tanh(2x) = 2\operatorname{sech}^2(2x).

Example 2. \frac{d}{dx}\tanh(x^2) = 2x\operatorname{sech}^2(x^2).

Example 3: a log. \frac{d}{dx}\ln(\cosh x) = \frac{\sinh x}{\cosh x} = \tanh x. So \int\tanh x\,dx = \ln(\cosh x) + C.

Example 4: a value. At x = 1, \tanh 1 \approx 0.7616, so the slope is 1 - 0.7616^2 \approx 0.420.

Where 1 − tanh² x shows up

In machine learning, \tanh is a common activation function. Training needs its derivative many times, and 1 - \tanh^2 x lets a program reuse the value of \tanh x it already computed instead of evaluating \cosh x.

Derivative of artanh x

For \lvert x\rvert < 1, set \tanh y = x. Then (1 - \tanh^2 y)\,y' = 1, so

Common mistakes

1. Writing \sec^2 x. That is the derivative of the circular tangent.
2. Writing 1 + \tanh^2 x. The hyperbolic version has a minus sign.
3. Forgetting the inside derivative. \frac{d}{dx}\tanh(3x) = 3\operatorname{sech}^2(3x).

The hyperbolic functions page covers \operatorname{sech}, \operatorname{csch} and \coth as well.