6. Derivatives of Inverse, Exponential, & Logarithmic Functions

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → ∞ (1 − coth x) / (1 − tanh x)

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

128. Derive the formula dy/dx = 1/(1+x²) for the derivative of y = arctan(x) by differentiating both sides of the equivalent equation tan(y)=x.

130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

25. y = sinh⁻¹(√x)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

29. y = (1 - t)coth⁻¹(√t)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

31. y = cos⁻¹(x) - x sech⁻¹(x)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

33. y = csch⁻¹(1/2)^θ

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

35. y = sinh⁻¹(tan x)

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

15. y = sin⁻¹√(1-u²), 0<u<1

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

17. y = ln(arccos(x))

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

19. y = t arctan(t) - 1/2 ln(t)

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

21. y = z arcsec(z) - √(z² - 1), z>1

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

23. y = arccsc(secθ), 0<θ<π/2

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

21. y=arccos(x²)