7–28. Derivatives Evaluate the following derivatives.
d/dx (sin (ln x))
7–28. Derivatives Evaluate the following derivatives.
d/dx (sin (ln x))
A calculator has a built-in sinh⁻¹ x function, but no csch⁻¹ x function. How do you evaluate csch⁻¹ 5 on such a calculator?
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