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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 #1121 : Partial Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=y^4z - 2ysin{(x^3)}cos{(z)}\\&\text{In the direction of }\overrightarrow{v}=(6,0,10).\end{align*}$

Possible Answers:

${10y^4 + 20ysin{(x^3)}sin{(z)} - 36x^2ycos{(x^3)}cos{(z)}}$

${70x^2cos{(x^3)}sin{(z)}}$

${0.86y^4 + 1.72ysin{(x^3)}sin{(z)} - 3.06x^2ycos{(x^3)}cos{(z)}}$

${46.6y^3z - 23.3sin{(x^3)}cos{(z)} + 11.7y^4 + 23.3ysin{(x^3)}sin{(z)} - 70x^2ycos{(x^3)}cos{(z)}}$

Correct answer:

${0.86y^4 + 1.72ysin{(x^3)}sin{(z)} - 3.06x^2ycos{(x^3)}cos{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(6)^2+(0)^2+(10)^2}=11.66\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{6}{11.66}=0.51\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{11.66}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{10}{11.66}=0.86\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.51)(-6x^2ycos{(x^3)}cos{(z)})+(0)(4y^3z - 2sin{(x^3)}cos{(z)})+(0.86)(y^4 + 2ysin{(x^3)}sin{(z)})\\&D_{\overrightarrow{u}}(x,y,z)=0.86y^4 + 1.72ysin{(x^3)}sin{(z)} - 3.06x^2ycos{(x^3)}cos{(z)}\end{align*}$

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Example Question #54 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=\frac{{(3e^{(y^2)})}}{z} + 2^{(y^2)}xsin{(z^3)}\\&\text{In the direction of }\overrightarrow{v}=(13,0,-17).\end{align*}$

Possible Answers:

${89\cdot2^{(y^2)}yz^2cos{(z^3)}}$

${21.4\cdot2^{(y^2)}sin{(z^3)} -\frac{ {(64.2e^{(y^2)})}}{z^2} +\frac{ {(128ye^{(y^2)})}}{z} + 29.7\cdot2^{(y^2)}xysin{(z^3)} + 64.2\cdot2^{(y^2)}xz^2cos{(z^3)}}$

${13\cdot2^{(y^2)}sin{(z^3)} +\frac{ {(51e^{(y^2)})}}{z^2} - 51\cdot2^{(y^2)}xz^2cos{(z^3)}}$

${0.61\cdot2^{(y^2)}sin{(z^3)} +\frac{ {(2.37e^{(y^2)})}}{z^2} - 2.37\cdot2^{(y^2)}xz^2cos{(z^3)}}$

Correct answer:

${0.61\cdot2^{(y^2)}sin{(z^3)} +\frac{ {(2.37e^{(y^2)})}}{z^2} - 2.37\cdot2^{(y^2)}xz^2cos{(z^3)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(13)^2+(0)^2+(-17)^2}=21.4\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{13}{21.4}=0.61\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{21.4}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-17}{21.4}=-0.79\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.61)(2^{(y^2)}sin{(z^3)})+(0)(\frac{{(6ye^{(y^2)})}}{z} + 1.39\cdot2^{(y^2)}xysin{(z^3)})+(-0.79)(3\cdot2^{(y^2)}xz^2cos{(z^3)} -\frac{ {(3e^{(y^2)})}}{z^2})\\&D_{\overrightarrow{u}}(x,y,z)=0.61\cdot2^{(y^2)}sin{(z^3)} +\frac{ {(2.37e^{(y^2)})}}{z^2} - 2.37\cdot2^{(y^2)}xz^2cos{(z^3)}\end{align*}$

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Example Question #55 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=- yz^3e^{(x)} - ln{(x^2)}sin{(z^3)}cos{(y)}\\&\text{In the direction of }\overrightarrow{v}=(14,0,-6).\end{align*}$

Possible Answers:

${18yz^2e^{(x)} -\frac{ {(28sin{(z^3)}cos{(y)})}}{x} - 14yz^3e^{(x)} + 18z^2cos{(z^3)}ln{(x^2)}cos{(y)}}$

${\frac{{(91.4z^2cos{(z^3)}sin{(y)})}}{x} - 45.7z^2e^{(x)}}$

${15.2ln{(x^2)}sin{(z^3)}sin{(y)} -\frac{ {(30.5sin{(z^3)}cos{(y)})}}{x} - 45.7yz^2e^{(x)} - 15.2yz^3e^{(x)} - 15.2z^3e^{(x)} - 45.7z^2cos{(z^3)}ln{(x^2)}cos{(y)}}$

${1.17yz^2e^{(x)} -\frac{ {(1.84sin{(z^3)}cos{(y)})}}{x} - 0.92yz^3e^{(x)} + 1.17z^2cos{(z^3)}ln{(x^2)}cos{(y)}}$

Correct answer:

${1.17yz^2e^{(x)} -\frac{ {(1.84sin{(z^3)}cos{(y)})}}{x} - 0.92yz^3e^{(x)} + 1.17z^2cos{(z^3)}ln{(x^2)}cos{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(14)^2+(0)^2+(-6)^2}=15.23\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{14}{15.23}=0.92\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{15.23}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-6}{15.23}=-0.39\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.92)(-\frac{ {(2sin{(z^3)}cos{(y)})}}{x} - yz^3e^{(x)})+(0)(ln{(x^2)}sin{(z^3)}sin{(y)} - z^3e^{(x)})+(-0.39)(- 3yz^2e^{(x)} - 3z^2cos{(z^3)}ln{(x^2)}cos{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=1.17yz^2e^{(x)} -\frac{ {(1.84sin{(z^3)}cos{(y)})}}{x} - 0.92yz^3e^{(x)} + 1.17z^2cos{(z^3)}ln{(x^2)}cos{(y)}\end{align*}$

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Example Question #1122 : Partial Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-yz^2\\&\text{In the direction of }\overrightarrow{v}=(0,16,-5).\end{align*}$

Possible Answers:

${10yz - 16z^2}$

${0.6yz - 0.95z^2}$

${- 33.5yz - 16.8z^2}$

Correct answer:

${0.6yz - 0.95z^2}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(16)^2+(-5)^2}=16.76\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{16.76}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{16}{16.76}=0.95\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-5}{16.76}=-0.3\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(0.0)+(0.95)(-1z^2)+(-0.3)(-2yz)\\&D_{\overrightarrow{u}}(x,y,z)=0.6yz - 0.95z^2\end{align*}$

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Example Question #1123 : Partial Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=zsin{(x)}\\&\text{In the direction of }\overrightarrow{v}=(0,-20,-19).\end{align*}$

Possible Answers:

{-0.69sin{(x)}}

{27.6sin{(x)} + 27.6zcos{(x)}}

{0}

{-19sin{(x)}}

Correct answer:

{-0.69sin{(x)}}

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(-20)^2+(-19)^2}=27.59\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{27.59}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-20}{27.59}=-0.72\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-19}{27.59}=-0.69\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(zcos{(x)})+(-0.72)(0.0)+(-0.69)(sin{(x)})\\&D_{\overrightarrow{u}}(x,y,z)=-0.69sin{(x)}\end{align*}$

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Example Question #61 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=z^3sin{(x^2)}sin{(y^3)}\\&\text{In the direction of }\overrightarrow{v}=(11,-17,0).\end{align*}$

Possible Answers:

${364xy^2z^2cos{(x^2)}cos{(y^3)}}$

${1.08xz^3cos{(x^2)}sin{(y^3)} - 2.52y^2z^3cos{(y^3)}sin{(x^2)}}$

${22xz^3cos{(x^2)}sin{(y^3)} - 51y^2z^3cos{(y^3)}sin{(x^2)}}$

${60.8z^2sin{(x^2)}sin{(y^3)} + 40.5xz^3cos{(x^2)}sin{(y^3)} + 60.8y^2z^3cos{(y^3)}sin{(x^2)}}$

Correct answer:

${1.08xz^3cos{(x^2)}sin{(y^3)} - 2.52y^2z^3cos{(y^3)}sin{(x^2)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(11)^2+(-17)^2+(0)^2}=20.25\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{11}{20.25}=0.54\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-17}{20.25}=-0.84\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{20.25}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.54)(2xz^3cos{(x^2)}sin{(y^3)})+(-0.84)(3y^2z^3cos{(y^3)}sin{(x^2)})+(0)(3z^2sin{(x^2)}sin{(y^3)})\\&D_{\overrightarrow{u}}(x,y,z)=1.08xz^3cos{(x^2)}sin{(y^3)} - 2.52y^2z^3cos{(y^3)}sin{(x^2)}\end{align*}$

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Example Question #62 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=x^3sin{(y^3)}sin{(z^3)}\\&\text{In the direction of }\overrightarrow{v}=(-6,0,15).\end{align*}$

Possible Answers:

${436x^2y^2z^2cos{(y^3)}cos{(z^3)}}$

${45x^3z^2cos{(z^3)}sin{(y^3)} - 18x^2sin{(y^3)}sin{(z^3)}}$

${48.5x^2sin{(y^3)}sin{(z^3)} + 48.5x^3y^2cos{(y^3)}sin{(z^3)} + 48.5x^3z^2cos{(z^3)}sin{(y^3)}}$

${2.79x^3z^2cos{(z^3)}sin{(y^3)} - 1.11x^2sin{(y^3)}sin{(z^3)}}$

Correct answer:

${2.79x^3z^2cos{(z^3)}sin{(y^3)} - 1.11x^2sin{(y^3)}sin{(z^3)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-6)^2+(0)^2+(15)^2}=16.16\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-6}{16.16}=-0.37\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{16.16}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{15}{16.16}=0.93\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.37)(3x^2sin{(y^3)}sin{(z^3)})+(0)(3x^3y^2cos{(y^3)}sin{(z^3)})+(0.93)(3x^3z^2cos{(z^3)}sin{(y^3)})\\&D_{\overrightarrow{u}}(x,y,z)=2.79x^3z^2cos{(z^3)}sin{(y^3)} - 1.11x^2sin{(y^3)}sin{(z^3)}\end{align*}$

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Example Question #63 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=x^3z^2sin{(y^2)}\\&\text{In the direction of }\overrightarrow{v}=(0,17,-3).\end{align*}$

Possible Answers:

${34x^3yz^2cos{(y^2)} - 6x^3zsin{(y^2)}}$

${51.8x^2z^2sin{(y^2)} + 34.5x^3zsin{(y^2)} + 34.5x^3yz^2cos{(y^2)}}$

${1.96x^3yz^2cos{(y^2)} - 0.34x^3zsin{(y^2)}}$

${207x^2yzcos{(y^2)}}$

Correct answer:

${1.96x^3yz^2cos{(y^2)} - 0.34x^3zsin{(y^2)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(0)^2+(17)^2+(-3)^2}=17.26\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{0}{17.26}=0\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{17}{17.26}=0.98\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-3}{17.26}=-0.17\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0)(3x^2z^2sin{(y^2)})+(0.98)(2x^3yz^2cos{(y^2)})+(-0.17)(2x^3zsin{(y^2)})\\&D_{\overrightarrow{u}}(x,y,z)=1.96x^3yz^2cos{(y^2)} - 0.34x^3zsin{(y^2)}\end{align*}$

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Example Question #64 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=y^3sin{(z)} - 2z^2cos{(y^3)}\\&\text{In the direction of }\overrightarrow{v}=(-8,0,-3).\end{align*}$

Possible Answers:

{0}

${8.54y^3cos{(z)} - 34.2zcos{(y^3)} + 25.6y^2sin{(z)} + 51.2y^2z^2sin{(y^3)}}$

${1.4zcos{(y^3)} - 0.35y^3cos{(z)}}$

${12zcos{(y^3)} - 3y^3cos{(z)}}$

Correct answer:

${1.4zcos{(y^3)} - 0.35y^3cos{(z)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(-8)^2+(0)^2+(-3)^2}=8.54\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{-8}{8.54}=-0.94\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{0}{8.54}=0\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{-3}{8.54}=-0.35\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(-0.94)(0.0)+(0)(3y^2sin{(z)} + 6y^2z^2sin{(y^3)})+(-0.35)(y^3cos{(z)} - 4zcos{(y^3)})\\&D_{\overrightarrow{u}}(x,y,z)=1.4zcos{(y^3)} - 0.35y^3cos{(z)}\end{align*}$

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Example Question #65 : Directional Derivatives

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-cos{(z^3)}e^{(y)}\\&\text{In the direction of }\overrightarrow{v}=(1,-11,0).\end{align*}$

Possible Answers:

{0}

${11cos{(z^3)}e^{(y)}}$

${33.2z^2sin{(z^3)}e^{(y)} - 11.1cos{(z^3)}e^{(y)}}$

${cos{(z^3)}e^{(y)}}$

Correct answer:

${cos{(z^3)}e^{(y)}}$

Explanation:

$\begin{align*}&\text{To find }D_{\overrightarrow{u}}(x,y,z)\\&\text{First make sure that }\overrightarrow{v}\\&\text{is in unit vector form. Find the magnitude:}\\&|\overrightarrow{v}|=\sqrt{x^2+y^2+z^2}=\sqrt{(1)^2+(-11)^2+(0)^2}=11.05\\&\text{Unit vector components are:}\\&u_x=\frac{v_x}{|\overrightarrow{v}|}=\frac{1}{11.05}=0.09\\&u_y=\frac{v_y}{|\overrightarrow{v}|}=\frac{-11}{11.05}=-1\\&u_z=\frac{v_z}{|\overrightarrow{v}|}=\frac{0}{11.05}=0\\&D_{\overrightarrow{u}}(x,y,z)=u_x\frac{df}{dx}+u_y\frac{df}{dy}+u_z\frac{df}{dz}\\&\text{Therefore, for our values:}\\&D_{\overrightarrow{u}}(x,y,z)=(0.09)(0.0)+(-1)(-1cos{(z^3)}e^{(y)})+(0)(3z^2sin{(z^3)}e^{(y)})\\&D_{\overrightarrow{u}}(x,y,z)=cos{(z^3)}e^{(y)}\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1121 : Partial Derivatives

Example Question #54 : Directional Derivatives

Example Question #55 : Directional Derivatives

Example Question #1122 : Partial Derivatives

Example Question #1123 : Partial Derivatives

Example Question #61 : Directional Derivatives

Example Question #62 : Directional Derivatives

Example Question #63 : Directional Derivatives

Example Question #64 : Directional Derivatives

Example Question #65 : Directional Derivatives

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