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Calculus 3 : Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #621 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zx}\\&\text{Where }f(x,y,z)=\frac{(5z^{2}sin(y^{2}))}{x^{2}}+\frac{ (2e^{(x^{2})}e^{(3z)}ln(y))}{7}\end{align*}$

Possible Answers:

${\frac{(6e^{(x^{2})}e^{(3z)}ln(y))}{7}-\frac{ (10z^{2}sin(y^{2}))}{x^{3}}+\frac{ (10zsin(y^{2}))}{x^{2}}+\frac{ (4xe^{(x^{2})}e^{(3z)}ln(y))}{7}}$

${\frac{(12xe^{(x^{2})}e^{(3z)}ln(y))}{7}-\frac{ (20zsin(y^{2}))}{x^{3}}}$

${-(\frac{(6e^{(x^{2})}e^{(3z)}ln(y))}{7}+\frac{ (10zsin(y^{2}))}{x^{2}})\cdot(\frac{(10z^{2}sin(y^{2}))}{x^{3}}-\frac{ (4xe^{(x^{2})}e^{(3z)}ln(y))}{7})}$

${-(\frac{(6e^{(x^{2})}e^{(3z)}ln(y))}{7}+\frac{ (10zsin(y^{2}))}{x^{2}})\cdot(\frac{(20zsin(y^{2}))}{x^{3}}-\frac{ (12xe^{(x^{2})}e^{(3z)}ln(y))}{7})}$

Correct answer:

${\frac{(12xe^{(x^{2})}e^{(3z)}ln(y))}{7}-\frac{ (20zsin(y^{2}))}{x^{3}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(5z^{2}sin(y^{2}))}{x^{2}}+\frac{ (2e^{(x^{2})}e^{(3z)}ln(y))}{7}\\&f_{z}=\frac{(6e^{(x^{2})}e^{(3z)}ln(y))}{7}+\frac{ (10zsin(y^{2}))}{x^{2}}\\&f_{zx}=\frac{(12xe^{(x^{2})}e^{(3z)}ln(y))}{7}-\frac{ (20zsin(y^{2}))}{x^{3}}\end{align*}$

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Example Question #622 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zz}\\&\text{Where }f(x,y,z)=\frac{(cos(z^{2})sin(y))}{x^{2}}- cos(y^{2})e^{(4z)}sin(3x + x^{2})\end{align*}$

Possible Answers:

${(4cos(y^{2})e^{(4z)}sin(3x + x^{2}) +\frac{ (2zsin(z^{2})sin(y))}{x^{2}})\cdot(16cos(y^{2})e^{(4z)}sin(3x + x^{2}) +\frac{ (2sin(z^{2})sin(y))}{x^{2}}+\frac{ (4z^{2}cos(z^{2})sin(y))}{x^{2}})}$

${- 16cos(y^{2})e^{(4z)}sin(3x + x^{2}) -\frac{ (2sin(z^{2})sin(y))}{x^{2}}-\frac{ (4z^{2}cos(z^{2})sin(y))}{x^{2}}}$

${- 8cos(y^{2})e^{(4z)}sin(3x + x^{2}) -\frac{ (4zsin(z^{2})sin(y))}{x^{2}}}$

${(4cos(y^{2})e^{(4z)}sin(3x + x^{2}) +\frac{ (2zsin(z^{2})sin(y))}{x^{2}})^{2}}$

Correct answer:

${- 16cos(y^{2})e^{(4z)}sin(3x + x^{2}) -\frac{ (2sin(z^{2})sin(y))}{x^{2}}-\frac{ (4z^{2}cos(z^{2})sin(y))}{x^{2}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(cos(z^{2})sin(y))}{x^{2}}- cos(y^{2})e^{(4z)}sin(3x + x^{2})\\&f_{z}=- 4cos(y^{2})e^{(4z)}sin(3x + x^{2}) -\frac{ (2zsin(z^{2})sin(y))}{x^{2}}\\&f_{zz}=- 16cos(y^{2})e^{(4z)}sin(3x + x^{2}) -\frac{ (2sin(z^{2})sin(y))}{x^{2}}-\frac{ (4z^{2}cos(z^{2})sin(y))}{x^{2}}\end{align*}$

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Example Question #623 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xz}\\&\text{Where }f(x,y,z)=\frac{(3cos(y))}{z}- 3ln(3z)e^{(x)}cos(3y + y^{2})\end{align*}$

Possible Answers:

${-\frac{(3e^{(x)}cos(3y + y^{2}))}{z}}$

${3ln(3z)e^{(x)}cos(3y + y^{2})\cdot(\frac{(3cos(y))}{z^{2}}+\frac{ (3e^{(x)}cos(3y + y^{2}))}{z})}$

${-\frac{ (3cos(y))}{z^{2}}-\frac{ (3e^{(x)}cos(3y + y^{2}))}{z}- 3ln(3z)e^{(x)}cos(3y + y^{2})}$

${\frac{(9ln(3z)e^{(2x)}cos(3y + y^{2})^{2})}{z}}$

Correct answer:

${-\frac{(3e^{(x)}cos(3y + y^{2}))}{z}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(3cos(y))}{z}- 3ln(3z)e^{(x)}cos(3y + y^{2})\\&f_{x}=-3ln(3z)e^{(x)}cos(3y + y^{2})\\&f_{xz}=-\frac{(3e^{(x)}cos(3y + y^{2}))}{z}\end{align*}$

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Example Question #624 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xx}\\&\text{Where }f(x,y,z)=3e^{(y)}sin(z) -\frac{ (2^{(4y)}\cdot4^{(x^{2})}ln(3z))}{2}\end{align*}$

Possible Answers:

${2^{(8y)}\cdot4^{(2x^{2})}x^{2}ln(3z)^{2}ln(4)^{2}}$

${- 2^{(4y)}\cdot4^{(x^{2})}ln(3z)ln(4) - 2\cdot2^{(4y)}\cdot4^{(x^{2})}x^{2}ln(3z)ln(4)^{2}}$

${2^{(4y)}\cdot4^{(x^{2})}xln(3z)ln(4)\cdot(2^{(4y)}\cdot4^{(x^{2})}ln(3z)ln(4) + 2\cdot2^{(4y)}\cdot4^{(x^{2})}x^{2}ln(3z)ln(4)^{2})}$

${-2\cdot2^{(4y)}\cdot4^{(x^{2})}xln(3z)ln(4)}$

Correct answer:

${- 2^{(4y)}\cdot4^{(x^{2})}ln(3z)ln(4) - 2\cdot2^{(4y)}\cdot4^{(x^{2})}x^{2}ln(3z)ln(4)^{2}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[a^u]=a^uduln(a)\\&\text{Solve step-by-step:}\\&f(x,y,z)=3e^{(y)}sin(z) -\frac{ (2^{(4y)}\cdot4^{(x^{2})}ln(3z))}{2}\\&f_{x}=-2^{(4y)}\cdot4^{(x^{2})}xln(3z)ln(4)\\&f_{xx}=- 2^{(4y)}\cdot4^{(x^{2})}ln(3z)ln(4) - 2\cdot2^{(4y)}\cdot4^{(x^{2})}x^{2}ln(3z)ln(4)^{2}\end{align*}$

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Example Question #625 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zx}\\&\text{Where }f(x,y,z)=2y^{4}sin(z) +\frac{ (3^{(4x)}cos(y^{2})ln(z))}{5}\end{align*}$

Possible Answers:

${\frac{(4\cdot3^{(4x)}cos(y^{2})ln(3)ln(z)\cdot(2y^{4}cos(z) +\frac{ (3^{(4x)}cos(y^{2}))}{(5z)}))}{5}}$

${2y^{4}cos(z) +\frac{ (3^{(4x)}cos(y^{2}))}{(5z)}+\frac{ (4\cdot3^{(4x)}cos(y^{2})ln(3)ln(z))}{5}}$

${\frac{(4\cdot3^{(4x)}cos(y^{2})ln(3)\cdot(2y^{4}cos(z) +\frac{ (3^{(4x)}cos(y^{2}))}{(5z)}))}{(5z)}}$

${\frac{(4\cdot3^{(4x)}cos(y^{2})ln(3))}{(5z)}}$

Correct answer:

${\frac{(4\cdot3^{(4x)}cos(y^{2})ln(3))}{(5z)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[a^u]=a^uduln(a)\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=2y^{4}sin(z) +\frac{ (3^{(4x)}cos(y^{2})ln(z))}{5}\\&f_{z}=2y^{4}cos(z) +\frac{ (3^{(4x)}cos(y^{2}))}{(5z)}\\&f_{zx}=\frac{(4\cdot3^{(4x)}cos(y^{2})ln(3))}{(5z)}\end{align*}$

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Example Question #626 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yx}\\&\text{Where }f(x,y,z)=-2\cdot2^{(4x)}ln(3z)cos(y^{2} - 4y)\end{align*}$

Possible Answers:

${2\cdot2^{(4x)}ln(3z)sin(y^{2} - 4y)\cdot(2y - 4) - 8\cdot2^{(4x)}ln(3z)ln(2)cos(y^{2} - 4y)}$

${-16\cdot2^{(8x)}ln(3z)^{2}ln(2)cos(y^{2} - 4y)sin(y^{2} - 4y)\cdot(2y - 4)}$

${8\cdot2^{(4x)}ln(3z)ln(2)sin(y^{2} - 4y)\cdot(2y - 4)}$

${16\cdot2^{(8x)}ln(3z)^{2}ln(2)sin(y^{2} - 4y)^{2}\cdot(2y - 4)^{2}}$

Correct answer:

${8\cdot2^{(4x)}ln(3z)ln(2)sin(y^{2} - 4y)\cdot(2y - 4)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[a^u]=a^uduln(a)\\&\text{Solve step-by-step:}\\&f(x,y,z)=-2\cdot2^{(4x)}ln(3z)cos(y^{2} - 4y)\\&f_{y}=2\cdot2^{(4x)}ln(3z)sin(y^{2} - 4y)\cdot(2y - 4)\\&f_{yx}=8\cdot2^{(4x)}ln(3z)ln(2)sin(y^{2} - 4y)\cdot(2y - 4)\end{align*}$

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Example Question #627 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yy}\\&\text{Where }f(x,y,z)=2sin(y^{2})ln(x)sin(z) - 4e^{(3y)}cos(x^{2} - 3x)cos(z)\end{align*}$

Possible Answers:

${8ycos(y^{2})ln(x)sin(z) - 24e^{(3y)}cos(x^{2} - 3x)cos(z)}$

${4cos(y^{2})ln(x)sin(z) - 36e^{(3y)}cos(x^{2} - 3x)cos(z) - 8y^{2}sin(y^{2})ln(x)sin(z)}$

${(12e^{(3y)}cos(x^{2} - 3x)cos(z) - 4ycos(y^{2})ln(x)sin(z))^{2}}$

${(12e^{(3y)}cos(x^{2} - 3x)cos(z) - 4ycos(y^{2})ln(x)sin(z))\cdot(36e^{(3y)}cos(x^{2} - 3x)cos(z) - 4cos(y^{2})ln(x)sin(z) + 8y^{2}sin(y^{2})ln(x)sin(z))}$

Correct answer:

${4cos(y^{2})ln(x)sin(z) - 36e^{(3y)}cos(x^{2} - 3x)cos(z) - 8y^{2}sin(y^{2})ln(x)sin(z)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=2sin(y^{2})ln(x)sin(z) - 4e^{(3y)}cos(x^{2} - 3x)cos(z)\\&f_{y}=4ycos(y^{2})ln(x)sin(z) - 12e^{(3y)}cos(x^{2} - 3x)cos(z)\\&f_{yy}=4cos(y^{2})ln(x)sin(z) - 36e^{(3y)}cos(x^{2} - 3x)cos(z) - 8y^{2}sin(y^{2})ln(x)sin(z)\end{align*}$

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Example Question #628 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xz}\\&\text{Where }f(x,y,z)=\frac{(ln(4x)ln(4z))}{y}- 4x^{3}cos(y)ln(z)\end{align*}$

Possible Answers:

${\frac{ln(4z)}{(xy)}+\frac{ ln(4x)}{(yz)}-\frac{ (4x^{3}cos(y))}{z}- 12x^{2}cos(y)ln(z)}$

${(\frac{ln(4x)}{(yz)}-\frac{ (4x^{3}cos(y))}{z})\cdot(\frac{ln(4z)}{(xy)}- 12x^{2}cos(y)ln(z))}$

${(\frac{1}{(xyz)}-\frac{ (12x^{2}cos(y))}{z})\cdot(\frac{ln(4z)}{(xy)}- 12x^{2}cos(y)ln(z))}$

${\frac{1}{(xyz)}-\frac{ (12x^{2}cos(y))}{z}}$

Correct answer:

${\frac{1}{(xyz)}-\frac{ (12x^{2}cos(y))}{z}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(ln(4x)ln(4z))}{y}- 4x^{3}cos(y)ln(z)\\&f_{x}=\frac{ln(4z)}{(xy)}- 12x^{2}cos(y)ln(z)\\&f_{xz}=\frac{1}{(xyz)}-\frac{ (12x^{2}cos(y))}{z}\end{align*}$

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Example Question #631 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xy}\\&\text{Where }f(x,y,z)=-3^{(z^{2})}sin(4x + x^{2})sin(y)\end{align*}$

Possible Answers:

${- 3^{(z^{2})}sin(4x + x^{2})cos(y) - 3^{(z^{2})}cos(4x + x^{2})sin(y)\cdot(2x + 4)}$

${3^{(2z^{2})}cos(4x + x^{2})sin(4x + x^{2})cos(y)sin(y)\cdot(2x + 4)}$

${-3^{(z^{2})}cos(4x + x^{2})cos(y)\cdot(2x + 4)}$

${3^{(2z^{2})}cos(4x + x^{2})^{2}cos(y)sin(y)\cdot(2x + 4)^{2}}$

Correct answer:

${-3^{(z^{2})}cos(4x + x^{2})cos(y)\cdot(2x + 4)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=-3^{(z^{2})}sin(4x + x^{2})sin(y)\\&f_{x}=-3^{(z^{2})}cos(4x + x^{2})sin(y)\cdot(2x + 4)\\&f_{xy}=-3^{(z^{2})}cos(4x + x^{2})cos(y)\cdot(2x + 4)\end{align*}$

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Example Question #632 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zz}\\&\text{Where }f(x,y,z)=\frac{(2\cdot2^{(3y)}cos(4x + x^{2})ln(z))}{7}- 4ln(y^{2})sin(z^{3})\end{align*}$

Possible Answers:

${(12z^{2}cos(z^{3})ln(y^{2}) -\frac{ (2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z)})\cdot(24zcos(z^{3})ln(y^{2}) - 36z^{4}ln(y^{2})sin(z^{3}) +\frac{ (2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z^{2})})}$

${36z^{4}ln(y^{2})sin(z^{3}) - 24zcos(z^{3})ln(y^{2}) -\frac{ (2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z^{2})}}$

${\frac{(4\cdot2^{(3y)}cos(4x + x^{2}))}{(7z)}- 24z^{2}cos(z^{3})ln(y^{2})}$

${(12z^{2}cos(z^{3})ln(y^{2}) -\frac{ (2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z)})^{2}}$

Correct answer:

${36z^{4}ln(y^{2})sin(z^{3}) - 24zcos(z^{3})ln(y^{2}) -\frac{ (2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z^{2})}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(2\cdot2^{(3y)}cos(4x + x^{2})ln(z))}{7}- 4ln(y^{2})sin(z^{3})\\&f_{z}=\frac{(2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z)}- 12z^{2}cos(z^{3})ln(y^{2})\\&f_{zz}=36z^{4}ln(y^{2})sin(z^{3}) - 24zcos(z^{3})ln(y^{2}) -\frac{ (2\cdot2^{(3y)}cos(4x + x^{2}))}{(7z^{2})}\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #621 : Partial Derivatives

Example Question #622 : Partial Derivatives

Example Question #623 : Partial Derivatives

Example Question #624 : Partial Derivatives

Example Question #625 : Partial Derivatives

Example Question #626 : Partial Derivatives

Example Question #627 : Partial Derivatives

Example Question #628 : Partial Derivatives

Example Question #631 : Partial Derivatives

Example Question #632 : Partial Derivatives

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