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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #501 : Partial Derivatives

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

Possible Answers:

${\frac{{(14x^4cos{(z^2)})}}{y^3} -\frac{ {(28x^3cos{(z^2)})}}{y^2} +\frac{ {(28x^4zsin{(z^2)})}}{y^2}}$

${-\frac{ {(112x^3sin{(z^2)})}}{y^3} -\frac{ {(224x^3z^2cos{(z^2)})}}{y^3}}$

$\frac{-{(76832x^{15}z^2cos{(z^2)}^2sin{(z^2)}^2)}}{y^9}$

$\frac{{(76832x^{15}z^2cos{(z^2)}^2sin{(z^2)}^2)}}{y^9}$

Correct answer:

${-\frac{ {(112x^3sin{(z^2)})}}{y^3} -\frac{ {(224x^3z^2cos{(z^2)})}}{y^3}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f_{z}=\frac{{(14x^4zsin{(z^2)})}}{y^2}\\&f_{zx}=\frac{{(56x^3zsin{(z^2)})}}{y^2}\\&f_{zxy}=\frac{-{(112x^3zsin{(z^2)})}}{y^3}\\&f_{zxyz}=-\frac{ {(112x^3sin{(z^2)})}}{y^3} -\frac{ {(224x^3z^2cos{(z^2)})}}{y^3}\end{align*}$

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Example Question #502 : Partial Derivatives

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

Possible Answers:

${-120y^4zcos{(x^2)}}$

$-8000y^{12}z^9cos{(x^2)}^3$

${-60y^4z^3cos{(x^2)}}$

${-144000y^{12}z^6cos{(x^2)}^3}$

Correct answer:

${-120y^4zcos{(x^2)}}$

Explanation:

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Example Question #503 : Partial Derivatives

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

Possible Answers:

$-16xzcos{(z^2)}sin{(z^2)}e^{(2x^2)}ln{(y)}^2$

${-8xzsin{(z^2)}e^{(x^2)}ln{(y)}}$

${32xz^2sin{(z^2)}^2e^{(2x^2)}ln{(y)}^2}$

${4xcos{(z^2)}e^{(x^2)}ln{(y)} - 4zsin{(z^2)}e^{(x^2)}ln{(y)}}$

Correct answer:

${-8xzsin{(z^2)}e^{(x^2)}ln{(y)}}$

Explanation:

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Example Question #504 : Partial Derivatives

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

Possible Answers:

$\frac{-{(2048\cdot4^{(3z^4)}ln{(y^4)})}}{{(x^7y^2)}}$

${\frac{{(32\cdot4^{(z^4)})}}{{(x^3y^2)}}}$

${\frac{{(32\cdot4^{(z^4)})}}{{(x^2y)}} -\frac{ {(8\cdot4^{(z^4)}ln{(y^4)})}}{x^3}}$

${\frac{-{(16384\cdot4^{(3z^4)})}}{{(x^8y^4)}}}$

Correct answer:

${\frac{{(32\cdot4^{(z^4)})}}{{(x^3y^2)}}}$

Explanation:

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Example Question #505 : Partial Derivatives

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

Possible Answers:

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

$\frac{{(1152y^9z^4cos{(z^3)}^2ln{(x^2)}^2sin{(z^3)})}}{x}$

$\frac{-{(1152y^9z^4cos{(z^3)}^2ln{(x^2)}^2sin{(z^3)})}}{x}$

${\frac{{(72y^3z^4sin{(z^3)})}}{x} -\frac{ {(48y^3zcos{(z^3)})}}{x}}$

Correct answer:

${\frac{{(72y^3z^4sin{(z^3)})}}{x} -\frac{ {(48y^3zcos{(z^3)})}}{x}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f_{x}=\frac{-{(8y^3sin{(z^3)})}}{x}\\&f_{xz}=\frac{-{(24y^3z^2cos{(z^3)})}}{x}\\&f_{xzz}=\frac{{(72y^3z^4sin{(z^3)})}}{x} -\frac{ {(48y^3zcos{(z^3)})}}{x}\end{align*}$

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Example Question #506 : Partial Derivatives

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

Possible Answers:

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

${\frac{-{(4zln{(x^2)}sin{(z^2)})}}{y^2}}$

${\frac{{(2cos{(z^2)}ln{(x^2)})}}{y^2} +\frac{ {(4zln{(x^2)}sin{(z^2)})}}{y}}$

$\frac{{(8zcos{(z^2)}ln{(x^2)}^2sin{(z^2)})}}{y^3}$

Correct answer:

${\frac{-{(4zln{(x^2)}sin{(z^2)})}}{y^2}}$

Explanation:

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Example Question #507 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zy}\\&\text{Where }f(x,y,z)=-7\cdot3^{(y^4)}sin{(z^4)}e^{(x^4)}\end{align*}$

Possible Answers:

$784\cdot3^{(2y^4)}y^3z^3cos{(z^4)}sin{(z^4)}e^{(2x^4)}ln{(3)}$

${3136\cdot3^{(2y^4)}y^3z^6cos{(z^4)}^2e^{(2x^4)}ln{(3)}}$

${-112\cdot3^{(y^4)}y^3z^3cos{(z^4)}e^{(x^4)}ln{(3)}}$

${- 28\cdot3^{(y^4)}z^3cos{(z^4)}e^{(x^4)} - 28\cdot3^{(y^4)}y^3sin{(z^4)}e^{(x^4)}ln{(3)}}$

Correct answer:

${-112\cdot3^{(y^4)}y^3z^3cos{(z^4)}e^{(x^4)}ln{(3)}}$

Explanation:

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Example Question #508 : Partial Derivatives

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

Possible Answers:

${54z^2sin{(x^2)}e^{(y^2)} + 18yz^3sin{(x^2)}e^{(y^2)}}$

${108yzsin{(x^2)}e^{(y^2)}}$

$13122yz^7sin{(x^2)}^3e^{(3y^2)}$

${104976y^3z^6sin{(x^2)}^3e^{(3y^2)}}$

Correct answer:

${108yzsin{(x^2)}e^{(y^2)}}$

Explanation:

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Example Question #509 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zxzz}\\&\text{Where }f(x,y,z)=10\cdot2^{(y^2)}cos{(x^4)}cos{(z^4)}\end{align*}$

Possible Answers:

${960\cdot2^{(y^2)}x^3zsin{(x^4)}sin{(z^4)} + 5760\cdot2^{(y^2)}x^3z^5cos{(z^4)}sin{(x^4)} - 2560\cdot2^{(y^2)}x^3z^9sin{(x^4)}sin{(z^4)}}$

${-6400\cdot2^{(2y^2)}x^3z^6cos{(x^4)}sin{(x^4)}sin{(z^4)}^2\cdot{(640\cdot2^{(y^2)}x^3z^6cos{(z^4)}sin{(x^4)} + 480\cdot2^{(y^2)}x^3z^2sin{(x^4)}sin{(z^4)})}\cdot{(960\cdot2^{(y^2)}x^3zsin{(x^4)}sin{(z^4)} + 5760\cdot2^{(y^2)}x^3z^5cos{(z^4)}sin{(x^4)} - 2560\cdot2^{(y^2)}x^3z^9sin{(x^4)}sin{(z^4)})}}$

$2560000\cdot2^{(4y^2)}x^3z^9cos{(x^4)}^3cos{(z^4)}sin{(x^4)}sin{(z^4)}^3$

${- 40\cdot2^{(y^2)}x^3cos{(z^4)}sin{(x^4)} - 120\cdot2^{(y^2)}z^3cos{(x^4)}sin{(z^4)}}$

Correct answer:

${960\cdot2^{(y^2)}x^3zsin{(x^4)}sin{(z^4)} + 5760\cdot2^{(y^2)}x^3z^5cos{(z^4)}sin{(x^4)} - 2560\cdot2^{(y^2)}x^3z^9sin{(x^4)}sin{(z^4)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f_{z}=-40\cdot2^{(y^2)}z^3cos{(x^4)}sin{(z^4)}\\&f_{zx}=160\cdot2^{(y^2)}x^3z^3sin{(x^4)}sin{(z^4)}\\&f_{zxz}=640\cdot2^{(y^2)}x^3z^6cos{(z^4)}sin{(x^4)} + 480\cdot2^{(y^2)}x^3z^2sin{(x^4)}sin{(z^4)}\\&f_{zxzz}=960\cdot2^{(y^2)}x^3zsin{(x^4)}sin{(z^4)} + 5760\cdot2^{(y^2)}x^3z^5cos{(z^4)}sin{(x^4)} - 2560\cdot2^{(y^2)}x^3z^9sin{(x^4)}sin{(z^4)}\end{align*}$

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Example Question #510 : Partial Derivatives

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

Possible Answers:

${\frac{{(192y^6cos{(z^4)}^2ln{(x^3)}sin{(y^4)}^2)}}{x}}$

${\frac{{(6cos{(y^4)}cos{(z^4)})}}{x} - 8y^3cos{(z^4)}ln{(x^3)}sin{(y^4)}}$

$\frac{-{(48y^3cos{(y^4)}cos{(z^4)}^2ln{(x^3)}sin{(y^4)})}}{x}$

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

Correct answer:

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

Explanation:

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #501 : Partial Derivatives

Example Question #502 : Partial Derivatives

Example Question #503 : Partial Derivatives

Example Question #504 : Partial Derivatives

Example Question #505 : Partial Derivatives

Example Question #506 : Partial Derivatives

Example Question #507 : Partial Derivatives

Example Question #508 : Partial Derivatives

Example Question #509 : Partial Derivatives

Example Question #510 : Partial Derivatives

All Calculus 3 Resources

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