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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #761 : Partial Derivatives

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

Possible Answers:

${\frac{(8xcos(y^{4} - 4y)\cdot (4y^{3} - 4))}{z^{2}}-\frac{ (6y^{2}cos(y^{3})cos(z))}{x^{2}}}$

${-(\frac{(6y^{2}cos(y^{3})cos(z))}{x^{2}}-\frac{ (8xcos(y^{4} - 4y)\cdot (4y^{3} - 4))}{z^{2}})\cdot (\frac{(8xsin(y^{4} - 4y))}{z^{2}}-\frac{ (2sin(y^{3})cos(z))}{x^{2}})}$

${\frac{(8xsin(y^{4} - 4y))}{z^{2}}-\frac{ (2sin(y^{3})cos(z))}{x^{2}}+\frac{ (4x^{2}cos(y^{4} - 4y)\cdot (4y^{3} - 4))}{z^{2}}+\frac{ (6y^{2}cos(y^{3})cos(z))}{x}}$

${(\frac{(4x^{2}cos(y^{4} - 4y)\cdot (4y^{3} - 4))}{z^{2}}+\frac{ (6y^{2}cos(y^{3})cos(z))}{x})\cdot (\frac{(8xsin(y^{4} - 4y))}{z^{2}}-\frac{ (2sin(y^{3})cos(z))}{x^{2}})}$

Correct answer:

${\frac{(8xcos(y^{4} - 4y)\cdot (4y^{3} - 4))}{z^{2}}-\frac{ (6y^{2}cos(y^{3})cos(z))}{x^{2}}}$

Explanation:

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Example Question #762 : Partial Derivatives

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

Possible Answers:

${- 6yln(x^{2})tan(z^{3}) - 9y^{2}z^{2}ln(x^{2})\cdot (tan(z^{3})^{2} + 1)}$

${54y^{3}z^{2}ln(x^{2})^{2}tan(z^{3})\cdot (tan(z^{3})^{2} + 1)}$

${-18yz^{2}ln(x^{2})\cdot (tan(z^{3})^{2} + 1)}$

${162y^{3}z^{4}ln(x^{2})^{2}\cdot (tan(z^{3})^{2} + 1)^{2}}$

Correct answer:

${-18yz^{2}ln(x^{2})\cdot (tan(z^{3})^{2} + 1)}$

Explanation:

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Example Question #761 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yx}\\&\text{Where }f(x,y,z)=-\frac{ (2sin(x^{2})sin(4y + y^{2})tan(11z + z^{4}))}{5}-\frac{ (2x^{4}z^{2})}{y}\end{align*}$

Possible Answers:

${-(\frac{(2x^{4}z^{2})}{y^{2}}-\frac{ (2sin(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5})\cdot (\frac{(8x^{3}z^{2})}{y}+\frac{ (4xcos(x^{2})sin(4y + y^{2})tan(11z + z^{4}))}{5})}$

${\frac{(2x^{4}z^{2})}{y^{2}}-\frac{ (8x^{3}z^{2})}{y}-\frac{ (4xcos(x^{2})sin(4y + y^{2})tan(11z + z^{4}))}{5}-\frac{ (2sin(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5}}$

${\frac{(8x^{3}z^{2})}{y^{2}}-\frac{ (4xcos(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5}}$

${(\frac{(2x^{4}z^{2})}{y^{2}}-\frac{ (2sin(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5})\cdot (\frac{(8x^{3}z^{2})}{y^{2}}-\frac{ (4xcos(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5})}$

Correct answer:

${\frac{(8x^{3}z^{2})}{y^{2}}-\frac{ (4xcos(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=-\frac{ (2sin(x^{2})sin(4y + y^{2})tan(11z + z^{4}))}{5}-\frac{ (2x^{4}z^{2})}{y}\\&f_{y}=\frac{(2x^{4}z^{2})}{y^{2}}-\frac{ (2sin(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5}\\&f_{yx}=\frac{(8x^{3}z^{2})}{y^{2}}-\frac{ (4xcos(x^{2})cos(4y + y^{2})tan(11z + z^{4})\cdot (2y + 4))}{5}\end{align*}$

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Example Question #763 : Partial Derivatives

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

Possible Answers:

${\frac{(16y^{4})}{(x^{3}z^{3})}-\frac{ (8xz^{2}sin(x^{2})e^{(y^{2})})}{5}}$

${-(\frac{(8y^{4})}{(x^{3}z^{3})}-\frac{ (4xz^{2}sin(x^{2})e^{(y^{2})})}{5})\cdot (\frac{(4z^{2}sin(x^{2})e^{(y^{2})})}{5}+\frac{ (24y^{4})}{(x^{4}z^{3})}+\frac{ (8x^{2}z^{2}cos(x^{2})e^{(y^{2})})}{5})}$

${-\frac{ (4z^{2}sin(x^{2})e^{(y^{2})})}{5}-\frac{ (24y^{4})}{(x^{4}z^{3})}-\frac{ (8x^{2}z^{2}cos(x^{2})e^{(y^{2})})}{5}}$

${(\frac{(8y^{4})}{(x^{3}z^{3})}-\frac{ (4xz^{2}sin(x^{2})e^{(y^{2})})}{5})^{2}}$

Correct answer:

${-\frac{ (4z^{2}sin(x^{2})e^{(y^{2})})}{5}-\frac{ (24y^{4})}{(x^{4}z^{3})}-\frac{ (8x^{2}z^{2}cos(x^{2})e^{(y^{2})})}{5}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(2z^{2}cos(x^{2})e^{(y^{2})})}{5}-\frac{ (4y^{4})}{(x^{2}z^{3})}\\&f_{x}=\frac{(8y^{4})}{(x^{3}z^{3})}-\frac{ (4xz^{2}sin(x^{2})e^{(y^{2})})}{5}\\&f_{xx}=-\frac{ (4z^{2}sin(x^{2})e^{(y^{2})})}{5}-\frac{ (24y^{4})}{(x^{4}z^{3})}-\frac{ (8x^{2}z^{2}cos(x^{2})e^{(y^{2})})}{5}\end{align*}$

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Example Question #764 : Partial Derivatives

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

Possible Answers:

${16\cdot 2^{(8y)}x^{4}e^{(2z)}ln(2)}$

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

${16\cdot 2^{(8y)}x^{4}e^{(2z)}ln(2)}$

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

Correct answer:

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

Explanation:

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Example Question #765 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yy}\\&\text{Where }f(x,y,z)=5sin(y^{2})sin(z) +\frac{ (cos(x^{3} - 3x)tan(12y + y^{4})sin(4z + z^{2}))}{2}\end{align*}$

Possible Answers:

${(10ycos(y^{2})sin(z) +\frac{ (cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12))}{2})^{2}}$

${10cos(y^{2})sin(z) - 20y^{2}sin(y^{2})sin(z) + 6y^{2}cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1) + cos(x^{3} - 3x)tan(12y + y^{4})sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12)^{2}}$

${(10ycos(y^{2})sin(z) +\frac{ (cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12))}{2})\cdot (10cos(y^{2})sin(z) - 20y^{2}sin(y^{2})sin(z) + 6y^{2}cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1) + cos(x^{3} - 3x)tan(12y + y^{4})sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12)^{2})}$

${20ycos(y^{2})sin(z) + cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12)}$

Correct answer:

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=5sin(y^{2})sin(z) +\frac{ (cos(x^{3} - 3x)tan(12y + y^{4})sin(4z + z^{2}))}{2}\\&f_{y}=10ycos(y^{2})sin(z) +\frac{ (cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12))}{2}\\&f_{yy}=10cos(y^{2})sin(z) - 20y^{2}sin(y^{2})sin(z) + 6y^{2}cos(x^{3} - 3x)sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1) + cos(x^{3} - 3x)tan(12y + y^{4})sin(4z + z^{2})\cdot (tan(12y + y^{4})^{2} + 1)\cdot (4y^{3} + 12)^{2}\end{align*}$

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Example Question #766 : Partial Derivatives

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

Possible Answers:

${\frac{(4x^{4}sin(z^{2}))}{y}-\frac{ (x^{2}e^{(y^{2})})}{z}- x^{2}yln(z^{2})e^{(y^{2})} + 8x^{4}zcos(z^{2})ln(y)}$

${-(\frac{(x^{2}e^{(y^{2})})}{z}- 8x^{4}zcos(z^{2})ln(y))\cdot (\frac{(4x^{4}sin(z^{2}))}{y}- x^{2}yln(z^{2})e^{(y^{2})})}$

${-(\frac{(x^{2}e^{(y^{2})})}{z}- 8x^{4}zcos(z^{2})ln(y))\cdot (\frac{(8x^{4}zcos(z^{2}))}{y}-\frac{ (2x^{2}ye^{(y^{2})})}{z})}$

${\frac{(8x^{4}zcos(z^{2}))}{y}-\frac{ (2x^{2}ye^{(y^{2})})}{z}}$

Correct answer:

${\frac{(8x^{4}zcos(z^{2}))}{y}-\frac{ (2x^{2}ye^{(y^{2})})}{z}}$

Explanation:

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Example Question #767 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zx}\\&\text{Where }f(x,y,z)=-2sin(y^{2})tan(12x + x^{4})tan(11z + z^{2})\end{align*}$

Possible Answers:

${4sin(y^{2})^{2}tan(12x + x^{4})\cdot (2z + 11)^{2}\cdot (tan(12x + x^{4})^{2} + 1)\cdot (tan(11z + z^{2})^{2} + 1)^{2}\cdot (4x^{3} + 12)}$

${-2sin(y^{2})\cdot (2z + 11)\cdot (tan(12x + x^{4})^{2} + 1)\cdot (tan(11z + z^{2})^{2} + 1)\cdot (4x^{3} + 12)}$

${4sin(y^{2})^{2}tan(12x + x^{4})tan(11z + z^{2})\cdot (2z + 11)\cdot (tan(12x + x^{4})^{2} + 1)\cdot (tan(11z + z^{2})^{2} + 1)\cdot (4x^{3} + 12)}$

${- 2sin(y^{2})tan(11z + z^{2})\cdot (tan(12x + x^{4})^{2} + 1)\cdot (4x^{3} + 12) - 2sin(y^{2})tan(12x + x^{4})\cdot (2z + 11)\cdot (tan(11z + z^{2})^{2} + 1)}$

Correct answer:

${-2sin(y^{2})\cdot (2z + 11)\cdot (tan(12x + x^{4})^{2} + 1)\cdot (tan(11z + z^{2})^{2} + 1)\cdot (4x^{3} + 12)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=-2sin(y^{2})tan(12x + x^{4})tan(11z + z^{2})\\&f_{z}=-2sin(y^{2})tan(12x + x^{4})\cdot (2z + 11)\cdot (tan(11z + z^{2})^{2} + 1)\\&f_{zx}=-2sin(y^{2})\cdot (2z + 11)\cdot (tan(12x + x^{4})^{2} + 1)\cdot (tan(11z + z^{2})^{2} + 1)\cdot (4x^{3} + 12)\end{align*}$

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Example Question #768 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyz}\\&\text{Where }f(x,y,z)=x^{3}sin(z^{2}) -\frac{ (2e^{(x^{2})}sin(y^{2} - 4y)sin(z^{4} - 4z))}{9}\end{align*}$

Possible Answers:

${\frac{(2e^{(x^{2})}cos(y^{2} - 4y)sin(z^{4} - 4z)\cdot (2y - 4)\cdot (4z^{3} - 4)^{2})}{9}-\frac{ (8z^{2}e^{(x^{2})}cos(y^{2} - 4y)cos(z^{4} - 4z)\cdot (2y - 4))}{3}}$

${-\frac{(2e^{(x^{2})}cos(y^{2} - 4y)sin(z^{4} - 4z)\cdot (2y - 4)\cdot (2x^{3}zcos(z^{2}) -\frac{ (2e^{(x^{2})}cos(z^{4} - 4z)sin(y^{2} - 4y)\cdot (4z^{3} - 4))}{9})^{2})}{9}}$

${\frac{(2e^{(x^{2})}cos(y^{2} - 4y)cos(z^{4} - 4z)\cdot (2y - 4)\cdot (2x^{3}zcos(z^{2}) -\frac{ (2e^{(x^{2})}cos(z^{4} - 4z)sin(y^{2} - 4y)\cdot (4z^{3} - 4))}{9})\cdot (4z^{3} - 4)\cdot (\frac{(8z^{2}e^{(x^{2})}cos(y^{2} - 4y)cos(z^{4} - 4z)\cdot (2y - 4))}{3}-\frac{ (2e^{(x^{2})}cos(y^{2} - 4y)sin(z^{4} - 4z)\cdot (2y - 4)\cdot (4z^{3} - 4)^{2})}{9}))}{9}}$

${4x^{3}zcos(z^{2}) -\frac{ (4e^{(x^{2})}cos(z^{4} - 4z)sin(y^{2} - 4y)\cdot (4z^{3} - 4))}{9}-\frac{ (2e^{(x^{2})}cos(y^{2} - 4y)sin(z^{4} - 4z)\cdot (2y - 4))}{9}}$

Correct answer:

${\frac{(2e^{(x^{2})}cos(y^{2} - 4y)sin(z^{4} - 4z)\cdot (2y - 4)\cdot (4z^{3} - 4)^{2})}{9}-\frac{ (8z^{2}e^{(x^{2})}cos(y^{2} - 4y)cos(z^{4} - 4z)\cdot (2y - 4))}{3}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=x^{3}sin(z^{2}) -\frac{ (2e^{(x^{2})}sin(y^{2} - 4y)sin(z^{4} - 4z))}{9}\\&f_{z}=2x^{3}zcos(z^{2}) -\frac{ (2e^{(x^{2})}cos(z^{4} - 4z)sin(y^{2} - 4y)\cdot (4z^{3} - 4))}{9}\\&f_{zy}=-\frac{(2e^{(x^{2})}cos(y^{2} - 4y)cos(z^{4} - 4z)\cdot (2y - 4)\cdot (4z^{3} - 4))}{9}\\&f_{zyz}=\frac{(2e^{(x^{2})}cos(y^{2} - 4y)sin(z^{4} - 4z)\cdot (2y - 4)\cdot (4z^{3} - 4)^{2})}{9}-\frac{ (8z^{2}e^{(x^{2})}cos(y^{2} - 4y)cos(z^{4} - 4z)\cdot (2y - 4))}{3}\end{align*}$

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Example Question #769 : Partial Derivatives

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

Possible Answers:

${6ln(4x)sin(4y + y^{2})sin(z)\cdot (2y + 4) - 6sin(x^{2})sin(z^{3})e^{(y)} - 6xcos(x^{2})sin(z^{3})e^{(y)} -\frac{ (3cos(4y + y^{2})sin(z))}{x}}$

${-(\frac{(3cos(4y + y^{2})sin(z))}{x}+ 6xcos(x^{2})sin(z^{3})e^{(y)})\cdot (3sin(x^{2})sin(z^{3})e^{(y)} - 3ln(4x)sin(4y + y^{2})sin(z)\cdot (2y + 4))^{2}}$

${(\frac{(3cos(4y + y^{2})sin(z))}{x}+ 6xcos(x^{2})sin(z^{3})e^{(y)})\cdot (6xcos(x^{2})sin(z^{3})e^{(y)} -\frac{ (3sin(4y + y^{2})sin(z)\cdot (2y + 4))}{x})\cdot (\frac{(6sin(4y + y^{2})sin(z))}{x}+\frac{ (3cos(4y + y^{2})sin(z)\cdot (2y + 4)^{2})}{x}- 6xcos(x^{2})sin(z^{3})e^{(y)})}$

${\frac{(6sin(4y + y^{2})sin(z))}{x}+\frac{ (3cos(4y + y^{2})sin(z)\cdot (2y + 4)^{2})}{x}- 6xcos(x^{2})sin(z^{3})e^{(y)}}$

Correct answer:

${\frac{(6sin(4y + y^{2})sin(z))}{x}+\frac{ (3cos(4y + y^{2})sin(z)\cdot (2y + 4)^{2})}{x}- 6xcos(x^{2})sin(z^{3})e^{(y)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=- 3ln(4x)cos(4y + y^{2})sin(z) - 3sin(x^{2})sin(z^{3})e^{(y)}\\&f_{x}=-\frac{ (3cos(4y + y^{2})sin(z))}{x}- 6xcos(x^{2})sin(z^{3})e^{(y)}\\&f_{xy}=\frac{(3sin(4y + y^{2})sin(z)\cdot (2y + 4))}{x}- 6xcos(x^{2})sin(z^{3})e^{(y)}\\&f_{xyy}=\frac{(6sin(4y + y^{2})sin(z))}{x}+\frac{ (3cos(4y + y^{2})sin(z)\cdot (2y + 4)^{2})}{x}- 6xcos(x^{2})sin(z^{3})e^{(y)}\end{align*}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #761 : Partial Derivatives

Example Question #762 : Partial Derivatives

Example Question #761 : Partial Derivatives

Example Question #763 : Partial Derivatives

Example Question #764 : Partial Derivatives

Example Question #765 : Partial Derivatives

Example Question #766 : Partial Derivatives

Example Question #767 : Partial Derivatives

Example Question #768 : Partial Derivatives

Example Question #769 : Partial Derivatives

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