6. Derivatives of Inverse, Exponential, & Logarithmic Functions

9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

y =√x³+x−1 at y=3

Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.

f(x) = x^-1/3

10–19. Derivatives Find the derivatives of the following functions.

f(t) = cosh t sinh t

10–19. Derivatives Find the derivatives of the following functions.

f(x) = tanh⁻¹(cos x)

10–19. Derivatives Find the derivatives of the following functions.

f(x) = (sinh x) / (1 + sinh x)

10–19. Derivatives Find the derivatives of the following functions.

g(t) = sinh⁻¹(√t)

Derivatives of hyperbolic functions Compute the following derivatives.

b. d/dx (x sech x)

Critical points Find the critical points of the function ƒ(x) = sinh² x cosh x.

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)