6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

7–28. Derivatives Evaluate the following derivatives.

d/dt ((sin t)^{√t})

7–28. Derivatives Evaluate the following derivatives.

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

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.)

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

c. How fast (in fish per year) is the population growing at t=0? At t=5?

Evaluate and simplify y'.

y = 4x⁴ ln x − x⁴

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

8. y = ln kx, k constant

9–61. Evaluate and simplify y'.

y = 2^x²−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.

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

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

7. y = log₂(x²/2)

15–48. Derivatives Find the derivative of the following functions.

f(x) = 2^x/2^x+1

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

15. y = 2√t tanh(√t)

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(cost+lnt)

63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

y = log₈ |tan x|

Find the derivative of the following functions.

y = In(cos² x)