6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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)

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

f(x) = x sinh⁻¹ x − √(x² + 1)

7–28. Derivatives Evaluate the following derivatives.

d/dx ((ln 2x)⁻⁵)

7–28. Derivatives Evaluate the following derivatives.

d/dx (ln³(3x² + 2))

7–28. Derivatives Evaluate the following derivatives.

d/dx ((2x)⁴ˣ)

7–28. Derivatives Evaluate the following derivatives.

d/dx (x^{π})

7–28. Derivatives Evaluate the following derivatives.

d/dt ((sin t)^{√t})

7–28. Derivatives Evaluate the following derivatives.

d/dy (y^{sin y})

7–28. Derivatives Evaluate the following derivatives.

d/dt (t^{1/t})

7–28. Derivatives Evaluate the following derivatives.

d/dx (e^{-10x²})

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

lim x → 0⁺ (tanh x)ˣ

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.

a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.

c. How long does it take for the BASE jumper to reach a speed of 45 m/s (roughly 100 mi/hr)?

Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).