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Calculus learning

Derivatives of Hyperbolic Functions: sinh, cosh and tanh

Hyperbolic derivatives can be derived directly from exponentials. This avoids transferring the wrong sign from the cosine rule. Expressing sinh and cosh through exp also provides runnable inputs for a parser without hyperbolic function names.

Direct answer

(sinh x)′=cosh x, (cosh x)′=sinh x, and (tanh x)′=1/cosh²x.

These three real functions are defined for every real x. Hyperbolic functions are not periodic trigonometric functions.

Derivatives of Hyperbolic Functions worked method

Define sinh x=(exp x−exp(−x))/2 and cosh x=(exp x+exp(−x))/2.

Differentiating exp(−x) contributes a minus sign. The difference in sinh becomes a sum, while the sum in cosh becomes a difference.

Write tanh x=sinh x/cosh x and apply the quotient rule. The numerator is cosh²x−sinh²x=1.

The denominator cosh²x is positive on the real line. For tanh(3x), multiply by 3 after preserving the inner argument.

Unlike cos, cosh never vanishes for real x. Reciprocal and inverse hyperbolic functions have their own domains and are outside this introductory lesson.

1
sinh x=(exp x−exp(−x))/2
2
cosh x=(exp x+exp(−x))/2
3
(tanh x)′=(cosh²x−sinh²x)/cosh²x

Worked examples

d/dx sinh(2x)
2cosh(2x)

The factor 2 comes from the chain rule.

d/dx cosh(x)
sinh(x)

No minus sign.

d/dx tanh(x)
1/cosh²(x)

Positive for real x.

Common mistakes

  • • Using (cosh x)′=−sinh x.
  • • Typing a literal function name unsupported by the parser.
  • • Confusing sinh⁻¹ with 1/sinh.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

This calculator example uses the exponential definition. Its result can be recognized as cosh(x), even when displayed with exp.

微分:ex−e−x2\frac{e^{x} - e^{-x}}{2}
Open the calculator without a preset: 微分

References

Frequently asked questions

Are these angle functions?+

They are defined through exponential functions; do not attach the trigonometric degree-conversion factor.

Can I use them in the current engine?+

Use their exp representations, as the preset does. Literal sinh/cosh/tanh aliases are not added in this release.

Does this cover inverse hyperbolic derivatives?+

No. Those require separate formulas and domain analysis.

Conclusion

Derivatives of Hyperbolic Functions starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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