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February 17th, 2015, 08:29 PM  #1 
Member Joined: Jan 2015 From: Colorado Posts: 32 Thanks: 0  Finding the rate of change at a point in the direction of a vector
Hi, I am confused on how to find the rate of change at a point in the direction of a vector, I know that when you are finding the rate of change at a point to another point, you take the change in distance for those two points, but I am not sure how to do that for this problem... What is the rate of change of f(x,y)=2xy+y2 at the point (3,2) in the direction v=−i+3j? 
February 18th, 2015, 03:49 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,462 Thanks: 2499 Math Focus: Mainly analysis and algebra 
This is a directional derivative. We are surfing along a line looking at how $f(x,y)$ changes along that line. In this case our line is $<3,2>+t<1,3> = <3t, 2+3t>$ where $t$ varies as we move along the line. And at $t=0$ we are at $<3,2>$. Thus we have the directional derivative$$f'(3,2) = \lim_{t \to 0} {f(3t,2+3t)  f(3,2) \over t}$$ This formula leads to $${\mathrm d \over \mathrm d t}f(x,y) = {\partial f \over \partial x}{\mathrm dx \over \mathrm d t} + {\partial f \over \partial x}{\mathrm dx \over \mathrm d t}$$ In our case $x = 3t$ and $y=2+3t$. 

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