Problem Set: Derivatives of Exponential and Logarithmic Functions

For the following exercises (1-15), find [latex]f^{\prime}(x)[/latex] for each function.

1. [latex]f(x)=x^2 e^x[/latex]

2. [latex]f(x)=\dfrac{e^{−x}}{x}[/latex]

3. [latex]f(x)=e^{x^3 \ln x}[/latex]

4. [latex]f(x)=\sqrt{e^{2x}+2x}[/latex]

5. [latex]f(x)=\dfrac{e^x-e^{−x}}{e^x+e^{−x}}[/latex]

6. [latex]f(x)=\dfrac{10^x}{\ln 10}[/latex]

7. [latex]f(x)=2^{4x}+4x^2[/latex]

8. [latex]f(x)=3^{\sin 3x}[/latex]

9. [latex]f(x)=x^{\pi} \cdot \pi^x[/latex]

10. [latex]f(x)=\ln(4x^3+x)[/latex]

11. [latex]f(x)=\ln \sqrt{5x-7}[/latex]

12. [latex]f(x)=x^2 \ln 9x[/latex]

13. [latex]f(x)=\log(\sec x)[/latex]

14. [latex]f(x)=\log_7 (6x^4+3)^5[/latex]

15. [latex]f(x)=2^x \cdot \log_3 7^{x^2-4}[/latex]

For the following exercises (16-23), use logarithmic differentiation to find [latex]\frac{dy}{dx}[/latex].

16. [latex]y=x^{\sqrt{x}}[/latex]

17. [latex]y=(\sin 2x)^{4x}[/latex]

18. [latex]y=(\ln x)^{\ln x}[/latex]

19. [latex]y=x^{\log_2 x}[/latex]

20. [latex]y=(x^2-1)^{\ln x}[/latex]

21. [latex]y=x^{\cot x}[/latex]

22. [latex]y= \dfrac{x+11}{\sqrt[3]{x^2-4}}[/latex]

23. [latex]y=x^{-\frac{1}{2}}(x^2+3)^{\frac{2}{3}}(3x-4)^4[/latex]

24. [T] Find an equation of the tangent line to the graph of [latex]f(x)=4xe^{x^2-1}[/latex] at the point where

[latex]x=-1[/latex]. Graph both the function and the tangent line.

25. [T] Find the equation of the line that is normal to the graph of [latex]f(x)=x \cdot 5^x[/latex] at the point where [latex]x=1[/latex]. Graph both the function and the normal line.

26. [T] Find the equation of the tangent line to the graph of [latex]x^3-x \ln y+y^3=2x+5[/latex] at the point where [latex]x=2[/latex]. Graph both the curve and the tangent line.

27. Consider the function [latex]y=x^{\frac{1}{x}}[/latex] for [latex]x>0[/latex].

  1. Determine the points on the graph where the tangent line is horizontal.
  2. Determine the intervals where [latex]y^{\prime}>0[/latex] and those where [latex]y^{\prime}<0[/latex].

28. The formula [latex]I(t)=\dfrac{\sin t}{e^t}[/latex] is the formula for a decaying alternating current.

  1. Complete the following table with the appropriate values.
    [latex]t[/latex] [latex]\frac{\sin t}{e^t}[/latex]
    0 (i)
    [latex]\frac{\pi}{2}[/latex] (ii)
    [latex]\pi[/latex] (iii)
    [latex]\frac{3\pi}{2}[/latex] (iv)
    [latex]2\pi[/latex] (v)
    [latex]2\pi[/latex] (vi)
    [latex]3\pi[/latex] (vii)
    [latex]\frac{7\pi}{2}[/latex] (viii)
    [latex]4\pi[/latex] (ix)
  2. Using only the values in the table, determine where the tangent line to the graph of [latex]I(t)[/latex] is horizontal.

29. [T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year.

  1. Write the exponential function that relates the total population as a function of [latex]t[/latex].
  2. Use a. to determine the rate at which the population is increasing in [latex]t[/latex] years.
  3. Use b. to determine the rate at which the population is increasing in 10 years.

30. [T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present.

  1. Write the exponential function that relates the amount of substance remaining as a function of [latex]t[/latex], measured in hours.
  2. Use a. to determine the rate at which the substance is decaying in [latex]t[/latex] hours.
  3. Use b. to determine the rate of decay at [latex]t=4[/latex] hours.

31. [T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1961 is modeled by the function

[latex]N(t)=5.3e^{0.093t^2-0.87t}, \, (0\le t\le 4)[/latex],

where [latex]N(t)[/latex] gives the number of cases (in thousands) and [latex]t[/latex] is measured in years, with [latex]t=0[/latex] corresponding to the beginning of 1960.

  1. Show work that evaluates [latex]N(0)[/latex] and [latex]N(4)[/latex]. Briefly describe what these values indicate about the disease in New York City.
  2. Show work that evaluates [latex]N^{\prime}(0)[/latex] and [latex]N^{\prime}(3)[/latex]. Briefly describe what these values indicate about the disease in New York City.

32. [T] The relative rate of change of a differentiable function [latex]y=f(x)[/latex] is given by [latex]\frac{100 \cdot f^{\prime}(x)}{f(x)}\%[/latex]. One model for population growth is a Gompertz growth function, given by [latex]P(x)=ae^{−b \cdot e^{−cx}}[/latex] where [latex]a, \, b[/latex], and [latex]c[/latex] are constants.

  1. Find the relative rate of change formula for the generic Gompertz function.
  2. Use a. to find the relative rate of change of a population in [latex]x=20[/latex] months when [latex]a=204,b=0.0198,[/latex] and [latex]c=0.15.[/latex]
  3. Briefly interpret what the result of b. means.

For the following exercises (33-36), use the population of New York City from 1790 to 1860, given in the following table.

New York City Population Over Time
Source: http://en.wikipedia.org/wiki/Largest_cities_in_the_United_States_by_population_by_decade
Years since 1790 Population
0 33,131
10 60,515
20 96,373
30 123,706
40 202,300
50 312,710
60 515,547
70 813,669

33. [T] Using a computer program or a calculator, fit a growth curve to the data of the form [latex]p=ab^t[/latex].

34. [T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.

35. [T] Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year.

36. [T] Using the tables of first and second derivatives and the best fit, answer the following questions:

  1. Will the model be accurate in predicting the future population of New York City? Why or why not?
  2. Estimate the population in 2010. Was the prediction correct from a.?