# Formulas for the Derivatives and Integrals of the Hyperbolic Functions | Summary and Q&A

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October 4, 2018
by
The Math Sorcerer
Formulas for the Derivatives and Integrals of the Hyperbolic Functions

## TL;DR

This video explains the derivatives and integrals of hyperbolic functions, providing formulas and examples.

## Key Insights

• ❓ Hyperbolic functions such as sinh, cosh, tanh, coth, sech, and csch have specific derivatives and integrals.
• ❓ The derivatives of hyperbolic functions closely resemble their trigonometric counterparts, with some variations.
• ❓ Integrating hyperbolic functions involves substituting variables and applying the appropriate formula.
• ❓ Memorizing the derivatives and integrals of hyperbolic functions simplifies problem-solving.
• 👨‍💼 The chain rule is used when differentiating functions involving hyperbolic sine.
• 💄 The integration of hyperbolic functions often requires making substitutions and applying the corresponding formula.
• 🤘 Differentiating hyperbolic secant involves adding a negative sign to the regular trigonometric derivative, secant tangent.
• 🤘 Integrating hyperbolic cosecant involves multiplying the regular trigonometric derivative, cosecant cotangent, by a negative sign.

## Transcript

hi YouTube in this video we're going to talk about the derivatives and integrals of the hyperbolic functions so first start with the derivative with respect to X of the hyperbolic sine of X cinch X so if you take this derivative it's actually really simple you just get the hyperbolic cosine of X totally worth memorizing this would mean that if you ... Read More

### Q: What is the derivative of hyperbolic sine (sinh) with respect to x?

The derivative of hyperbolic sine (sinh) with respect to x is hyperbolic cosine (cosh). This relationship is helpful when differentiating functions involving hyperbolic sine.

### Q: What is the integral of hyperbolic tangent (tanh) with respect to x?

The integral of hyperbolic tangent (tanh) with respect to x is hyperbolic secant squared (sech²) plus a constant. This formula allows for finding the integral of functions involving hyperbolic tangent.

### Q: How does the derivative of hyperbolic cosine (cosh) differ from regular trigonometric functions?

The derivative of hyperbolic cosine (cosh) with respect to x is hyperbolic sine (sinh), unlike the negative sine in regular trigonometric functions. This difference makes differentiating hyperbolic cosine simpler.

### Q: What is the integral of hyperbolic cotangent (coth) with respect to x?

The integral of hyperbolic cotangent (coth) with respect to x is negative hyperbolic cosecant squared (csch²) plus a constant. By knowing this, one can find the integral of functions involving hyperbolic cotangent.

## Summary & Key Takeaways

• The derivative of hyperbolic sine (sinh) with respect to x is hyperbolic cosine (cosh), and its integral is hyperbolic sine (sinh) plus a constant.

• The derivative of hyperbolic cosine (cosh) with respect to x is hyperbolic sine (sinh), and its integral is hyperbolic cosine (cosh) plus a constant.

• The derivative of hyperbolic tangent (tanh) with respect to x is hyperbolic secant squared (sech²), and its integral is hyperbolic tangent (tanh) plus a constant.

• The derivative of hyperbolic cotangent (coth) with respect to x is negative hyperbolic cosecant squared (csch²), and its integral is negative hyperbolic cotangent (coth) plus a constant.

• The derivative of hyperbolic secant (sech) with respect to x is negative hyperbolic secant (sech) hyperbolic tangent (tanh), and its integral is negative hyperbolic secant (sech) plus a constant.

• The derivative of hyperbolic cosecant (csch) with respect to x is negative hyperbolic cosecant (csch) hyperbolic cotangent (coth), and its integral is negative hyperbolic cosecant (csch) plus a constant.