Calculus Volume 1 · 3.9 Derivatives of Exponential and Logarithmic Functions

Derivatives of Exponential and Logarithmic Functions: practice problems

18 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 331

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=x^2{e}^x$

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2. Exercises 333

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=e^{x^3\ln x}$

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3. Exercises 335

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=\frac{e^x-e^{\text{-}x}}{e^x+e^{\text{-}x}}$

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4. Exercises 337

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=2^{4x}+4x^2$

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5. Exercises 339

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=x^{\pi }\cdot {\pi }^x$

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6. Exercises 341

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=\ln \sqrt{5x-7}$

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7. Exercises 343

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=\log (\sec x)$

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8. Exercises 345

   For the following exercises, find $f^{\prime }(x)$ for each function.

   $f(x)=2^x\cdot {\log }_3{7}^{x^2-4}$

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9. Exercises 347

   For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

   $y={(\sin 2x)}^{4x}$

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10. Exercises 349

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

    $y=x^{{\log }_{2}x}$

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11. Exercises 351

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

    $y=x^{\cot x}$

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12. Exercises 353

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

    $y=x^{-1\text{/}2}{(x^2+3)}^{2\text{/}3}{(3x-4)}^4$

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13. Exercises 355

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

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

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14. Exercises 357

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

    Consider the function $y=x^{1\text{/}x}$ for $x>0.$

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15. Exercises 359

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

    [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 $t.$
    2. Use a. to determine the rate at which the population is increasing in $t$ years.
    3. Use b. to determine the rate at which the population is increasing in 10 years.

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16. Exercises 361

    For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

    [T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1964 is modeled by the function

    $N(t)=5.3e^{0.093t^2-0.87t},(0\le t\le 4),$

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17. Exercises 363

    For the following exercises, use the population of New York City from 1790 to 1860, given in the following table.

    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

    [T] Using a computer program or a calculator, fit a growth curve to the data of the form $p=a{b}^t.$

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18. Exercises 365

    For the following exercises, use the population of New York City from 1790 to 1860, given in the following table.

    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

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

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