6. Derivatives of Inverse, Exponential, & 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).

f(x) = In(3x + 1)⁴

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 2x/(x² + 1)³

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 (2x - 1)(x + 2)³ / (1 - 4x)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).

f(x) = In(sec⁴x tan² x)

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

Use differentiation to verify each equation.

d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).

Evaluate and simplify y'.

y = 4x⁴ ln x − x⁴

Evaluate and simplify y'.

y = ln w / w⁵

9–61. Evaluate and simplify y'.

y = 2^x²−x

Evaluate and simplify y'.

y = ln |sec 3x|

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. 

Find d/dx(ln(x/x²+1)) without using the Quotient Rule.

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?

7–28. Derivatives Evaluate the following derivatives.

d/dx (sin (ln x))

A calculator has a built-in sinh⁻¹ x function, but no csch⁻¹ x function. How do you evaluate csch⁻¹ 5 on such a calculator?