6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.

Express sinh⁻¹ x in terms of logarithms.

On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?

What is the domain of sech⁻¹ x? How is sech⁻¹ x defined in terms of the inverse hyperbolic cosine?

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

cosh 2x = cosh²x + sinh²x (Hint: Begin with the right side of the equation.)

16–18. Identities Use the given identity to prove the related identity.

Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.

22–36. Derivatives Find the derivatives of the following functions.

f(x) = sinh 4x

22–36. Derivatives Find the derivatives of the following functions.

f(x) = cosh²x

22–36. Derivatives Find the derivatives of the following functions.

f(x) = tanh²x

22–36. Derivatives Find the derivatives of the following functions.

f(x) = √coth 3x

22–36. Derivatives Find the derivatives of the following functions.

f(x) = ln sech x

22–36. Derivatives Find the derivatives of the following functions.

f(x) = x² cosh² 3x

22–36. Derivatives Find the derivatives of the following functions.

f(t) = 2 tanh⁻¹ √t

22–36. Derivatives Find the derivatives of the following functions.

f(x) = csch⁻¹(2/x)