document.write( "Question 1106994: Let
\n" ); document.write( "f(x,y)=(x^2+2y^2)/
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\n" ); document.write( "Then the directional derivative of f at (0, 0) along u = (1, 1) is \n" ); document.write( "

Algebra.Com's Answer #722171 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "If \"s\" is the parameter on the straight line x = y along the vector (1,1), then we have  \r\n" );
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document.write( "    s = , y = ,   or, equivalently,  x = ,  y = .\r\n" );
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document.write( "Therefore, the numerator is  +  =  +  =  +  = ,  \r\n" );
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document.write( "while the denominator is x + y =  +  = .\r\n" );
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document.write( "Then the ratio itself is\r\n" );
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document.write( "f(x,y) = f(s) =  = .\r\n" );
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document.write( "Thus the function f(s) is LINEAR on s along this direction, and is zero at  x= y= 0= s by the definition, which is consistent with the linear behavior.\r\n" );
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document.write( "So (and therefore), the derivative    DOES EXIST  and is equal to .\r\n" );
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\n" ); document.write( "To avoid misunderstanding, let me note (highlight/underline) that for the given function the derivative \"along a direction\"
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\n" ); document.write( "\n" ); document.write( "It is ONLY differentiate \"along a direction\", and is a classic example of such a function.\r
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