6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Higher-order derivatives Find and simplify y''.

y = 2^x x

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.

d. Graph P' and use the graph to estimate the year in which the population is growing fastest. 

The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

Find the derivative of the given function.

$g(x)=e^{x}+\(\ln\) x^5$

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).

f(x) = In(3x + 1)⁴

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

lim x → 0⁺ (tanh x)ˣ

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

y = cos(e^(-θ^2))

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 = 4 log₃(x²−1)

Find the derivative of the given function.

$y=x^2\(\ln\)(x^2)$

Find the derivative of the following functions.

y = x² (1 - In x²)

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

12. y = ln(10/x)

Find the derivative of the given function.

$y=(4x-3x^2+9)\(\cdot\)2^{5x}$

Identify the open intervals on which the function is increasing or decreasing.

$f(x)=xe^{-2x}$

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).