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Find the directional derivative of f(x,y) = 3xy^{2} -4x^{3}y at (1,2) in the direction of...

Question:

Find the directional derivative of {eq}f(x,y) = 3xy^{2} -4x^{3}y {/eq} at (1,2) in the direction of {eq}\textbf{u} = 3\textbf{i} + 4\textbf{j} {/eq}

Directional Derivative:

The directional derivative {eq}\displaystyle{ \nabla _ { u } f ( x _ { 0 } , y _ { 0 } ) }{/eq} is the rate at which the function {eq}\displaystyle{ f ( x , y ) }{/eq} changes at a point {eq}\displaystyle{ (x_0,y_0) }{/eq} in the direction {eq}\displaystyle{ \mathbf u. }{/eq} It is a vector form of the usual derivative, and can be defined as

{eq}\displaystyle{ \nabla _ { u } f \equiv \nabla f \cdot \frac { u } { | u | }. }{/eq} The first step in taking a directional derivative, is to specify the direction. It can be specified along a vector {eq}\displaystyle{ \mathbf u. }{/eq}

Answer and Explanation:

We have,

{eq}\displaystyle{ f(x,y) = 3xy^{2} -4x^{3}y }{/eq}

The partial derivative of the functions can be defined as:

{eq}\displaystyle{ \nabl...

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Directional Derivatives, Gradient of f and the Min-Max

from GRE Math: Study Guide & Test Prep

Chapter 14 / Lesson 6
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