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Calculus 2 : First and Second Derivatives of Functions

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Example Questions

Example Question #431 : Derivative Review

$\begin{align*}&\text{Calculate the second derivative of the function}\\&17e^{(19x)}\end{align*}$

Possible Answers:

$1982251e^{(38x)}$

$104329e^{(38x)}$

$323e^{(19x)}$

$6137e^{(19x)}$

Correct answer:

$6137e^{(19x)}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&323e^{(19x)}\\&\text{And our second is:}\\&6137e^{(19x)}\end{align*}$

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Example Question #432 : Derivative Review

$\begin{align*}&\text{Calculate the second derivative of the function}\\&-3e^{(-13x)}\end{align*}$

Possible Answers:

$39e^{(-13x)}$

$-507e^{(-13x)}$

$1521e^{(-26x)}$

$-19773e^{(-26x)}$

Correct answer:

$-507e^{(-13x)}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&39e^{(-13x)}\\&\text{And our second is:}\\&-507e^{(-13x)}\end{align*}$

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Example Question #433 : Derivative Review

$\begin{align*}&\text{Calculate the second derivative of the function}\\&13cos(11x)\end{align*}$

Possible Answers:

-143cos(11x)

$20449sin(11x)^{2}$

224939cos(11x)sin(11x)

-1573cos(11x)

Correct answer:

-1573cos(11x)

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[cos(ax)]=-asin(ax)dx\\&d[sin(ax)]=acos(ax)dx\\&\text{Our first derivative is thus:}\\&-143sin(11x)\\&\text{And our second is:}\\&-1573cos(11x)\end{align*}$

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Example Question #434 : Derivative Review

$\begin{align*}&\text{Calculate the second derivative of the function}\\&3e^{(6x)}\end{align*}$

Possible Answers:

$1944e^{(12x)}$

$18e^{(6x)}$

$324e^{(12x)}$

$108e^{(6x)}$

Correct answer:

$108e^{(6x)}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Our first derivative is thus:}\\&18e^{(6x)}\\&\text{And our second is:}\\&108e^{(6x)}\end{align*}$

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Example Question #435 : Derivative Review

$\begin{align*}&\text{Take the derivative twice with respect to }x\text{ of the function}\\&y=19tan(11x)\end{align*}$

Possible Answers:

$418tan(11x)\cdot (11tan(11x)^{2} + 11)\cdot (209tan(11x)^{2} + 209)$

$38tan(11x)\cdot (11tan(11x)^{2} + 11)$

$418tan(11x)\cdot (11tan(11x)^{2} + 11)$

$(209tan(11x)^{2} + 209)^{2}$

Correct answer:

$418tan(11x)\cdot (11tan(11x)^{2} + 11)$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[tan(ax)]=\frac{adx}{cos^2(ax)}=(atan^2(ax)+a)dx\\&\text{Our first derivative is thus:}\\&209tan(11x)^{2} + 209\\&\text{And our second is:}\\&418tan(11x)\cdot (11tan(11x)^{2} + 11)\end{align*}$

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Example Question #436 : Derivative Review

$\begin{align*}&\text{Calculate the second derivative of the function}\\&-3\cdot 3^{(9x)}\end{align*}$

Possible Answers:

$6561\cdot 3^{(18x)}ln(3)^{3}$

$729\cdot 3^{(18x)}ln(3)^{2}$

$-243\cdot 3^{(9x)}ln(3)^{2}$

$-27\cdot 3^{(9x)}ln(3)^{2}$

Correct answer:

$-243\cdot 3^{(9x)}ln(3)^{2}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&\text{Our first derivative is thus:}\\&-27\cdot 3^{(9x)}ln(3)\\&\text{And our second is:}\\&-243\cdot 3^{(9x)}ln(3)^{2}\end{align*}$

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Example Question #437 : Derivative Review

$\begin{align*}&\text{Calculate the second derivative of the function}\\&-19x^{12}\end{align*}$

Possible Answers:

$-209x^{10}$

$571824x^{21}$

$51984x^{22}$

$-2508x^{10}$

Correct answer:

$-2508x^{10}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[x^a]=ax^{a-1}dx\\&\text{Our first derivative is thus:}\\&-228x^{11}\\&\text{And our second is:}\\&-2508x^{10}\end{align*}$

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Example Question #438 : Derivative Review

$\begin{align*}&\text{Take the derivative twice with respect to }x\text{ of the function}\\&y=17\cdot 4^{(2x)}\end{align*}$

Possible Answers:

$2312\cdot 4^{(4x)}ln(4)^{3}$

$1156\cdot 4^{(4x)}ln(4)^{2}$

$68\cdot 4^{(2x)}ln(4)^{2}$

$34\cdot 4^{(2x)}ln(4)^{2}$

Correct answer:

$68\cdot 4^{(2x)}ln(4)^{2}$

Explanation:

$\begin{align*}&\text{To find the second derivative of a function, one need only be}\\&\text{cautious in performing the steps of derivation twice. For the}\\&\text{function we're given,}\text{ we'll wish to use the following:}\\&d[b^{ax}]=ab^{ax}ln(b)sx\\&\text{Our first derivative is thus:}\\&34\cdot 4^{(2x)}ln(4)\\&\text{And our second is:}\\&68\cdot 4^{(2x)}ln(4)^{2}\end{align*}$

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Example Question #439 : Derivative Review

$\begin{align*}&\text{Take the derivative with respect to }x\text{ of the function}\\&y=130ln(9x)e^{(-12x)}\end{align*}$

Possible Answers:

$\frac{(120e^{(-12x)})}{x}+ 1404ln(9x)e^{(-12x)}$

$\frac{1}{x}- 12e^{(-12x)}$

$-\frac{(12e^{(-12x)})}{x}$

$\frac{(130e^{(-12x)})}{x}- 1560ln(9x)e^{(-12x)}$

Correct answer:

$\frac{(130e^{(-12x)})}{x}- 1560ln(9x)e^{(-12x)}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions multiplied together,}\\&\text{the first being: }e^{(-12x)}\\&\text{and the second being: }ln(9x)\\&\text{With the product scaled by }130\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[e^{ax}]=ae^{ax}dx\\&d[ln(ax)]=\frac{dx}{x}\\&u=e^{(-12x)};du=-12e^{(-12x)}\\&v=ln(9x);dv=\frac{1}{x}\\&\text{So the derivative is:}\\&\frac{(130e^{(-12x)})}{x}- 1560ln(9x)e^{(-12x)}\end{align*}$

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Example Question #440 : Derivative Review

$\begin{align*}&\text{Calculate the derivative of the function}\\&126x^{17}e^{(-4x)}\end{align*}$

Possible Answers:

$-68x^{16}e^{(-4x)}$

$2142x^{16}e^{(-4x)} - 504x^{17}e^{(-4x)}$

$476x^{16}e^{(-4x)} + 1224x^{17}e^{(-4x)}$

$17x^{16} - 4e^{(-4x)}$

Correct answer:

$2142x^{16}e^{(-4x)} - 504x^{17}e^{(-4x)}$

Explanation:

$\begin{align*}&\text{The function that we have been given can be seen as two smaller functions}\\&\text{multiplied together, the first being: }x^{17}\\&\text{and the second being: }e^{(-4x)}\\&\text{With the product scaled by }126\\&\text{To perform the derivative, }\text{ we'll wish to use the product rule, }d[uv]=udv+vdu\\&\text{along with the following:}\\&d[x^a]=ax^{a-1}dx\\&d[e^{ax}]=ae^{ax}dx\\&\text{From the above we can continue:}\\&u=x^{17};du=17x^{16}\\&v=e^{(-4x)};dv=-4e^{(-4x)}\\&\text{We can then combine these terms (Don't forget the scalar!)}\\&\text{So the derivative is:}\\&2142x^{16}e^{(-4x)} - 504x^{17}e^{(-4x)}\end{align*}$

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Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #431 : Derivative Review

Example Question #432 : Derivative Review

Example Question #433 : Derivative Review

Example Question #434 : Derivative Review

Example Question #435 : Derivative Review

Example Question #436 : Derivative Review

Example Question #437 : Derivative Review

Example Question #438 : Derivative Review

Example Question #439 : Derivative Review

Example Question #440 : Derivative Review

All Calculus 2 Resources

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