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Chapter 2: Q. 54 (page 222)

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = {(2 x + 1)}^{3 x}$

Short Answer

Expert verified

The derivative of function is $y' = {(2 x + 1)}^{3 x} [\frac{6 x}{2 x + 1} + \frac{3 \ln (2 x + 1)}{x}]$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = {(2 x + 1)}^{3 x}$

02

Step 2. Calculation

First, we will take the log on both sides and then differentiate it,

$\ln y = \ln {(2 x + 1)}^{3 x} \ln y = 3 x \cdot \ln (2 x + 1) \frac{1}{y} y' = \frac{3 x}{2 x + 1} (2) + \frac{\ln (2 x + 1)}{x} (3) y' = {(2 x + 1)}^{3 x} [\frac{6 x}{2 x + 1} + \frac{3 \ln (2 x + 1)}{x}]$

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Most popular questions from this chapter

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.The line tangent to the graph of $y = 1 - x - x^{2}$ at the point $(1, - 1)$

Find the derivative of the absolute value function and piecewise defined function

$f (x) = \{\begin{cases} 3 x + 1 & i f x \leq 1 \\ x^{3} & i f x > 1 \end{cases}$

Use (a) the $h \to 0$ definition of the derivative and then

(b) the $z \to c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23–38.

29. $f (x) = x^{\frac{1}{2}}, x = 9$

For each function f that follows find all the x-values in the domain of f for which $f' (x) = 0$ and all the values for which $f' (x)$ does not exist in later section we will call these values the critical points of f

localid="1648604345877" $a) f (x) = x^{3} - 2 x b) f (x) = \sqrt{x} - x c) f (x) = \frac{1}{1 + \sqrt{x}} d) f (x) = \frac{x^{2} (x - 1)}{{(x - 2)}^{2}}$

The total yearly expenditures by public colleges and universities from 1990 to 2000 can be modeled by the function $E (t) = 123 {(1.025)}^{t}$ , where expenditures are measured in billions of dollars and time is measured in years since 1990.

(a) Estimate the total yearly expenditures by these colleges and universities in 1995.

(b) Compute the average rate of change in yearly expenditures between 1990 and 2000.

(c) Compute the average rate of change in yearly expenditures between 1995 and 1996.

(d) Estimate the rate at which yearly expenditures of public colleges and universities were increasing in 1995.

See all solutions

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