What Are the Derivatives and Integrals of Hyperbolic Functions?

TL;DR

The derivatives and integrals of hyperbolic functions are closely related to their trigonometric counterparts. For instance, the derivative of hyperbolic sine (sinh) is hyperbolic cosine (cosh), and its integral is sinh plus a constant. Similarly, the derivative of hyperbolic tangent (tanh) is hyperbolic secant squared (sech²), while its integral is tanh plus a constant.

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

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.

Questions & Answers

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.