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Understand the problem

Let $\phi (x,y,z) = 3y^2+3yz$

for $(x,y,z) \in \mathbb{R}^{3}$

Then the absolute value of the directional derivative of $\phi$ in the direction of the line $\frac{x-1}{2} = \frac{y-2}{-1}= \frac{z}{-2}$

, at the point $(1,-2,1)$ is _____

Source of the problem

IIT JAM 2018

Topic

Directional Derivative

Difficulty Level

Easy

Suggested Book

Generalized Directional Derivatives By Pastor Karel

Start with hints

Do you really need a hint? Try it first!

The rate of change of $f(x,y,z)$ in the direction of the unit vector $\vec u = <a,b,c>$

is called the directional derivative and is denoted by $D_{\vec u}f(x,y,z)$

. The defination of directional derivative is $D_{\vec u}f(x,y,z) = \lim h \to 0 \frac{f(x+ah,y+bh,z+ch)-f(x,y)}{h}$

. Here if you consider $u$ to be $(1,0,0), (0,1,0)$

and $(0,0,1)$ then we will get $D_{\vec u}f(x,y,z)$

to be $f_x$ , $f_y$ and $f_z$ respectively. If you observe closely, this is nothing but differentiating the function $f$ w.r.t $x,y$ and $z$ keeping all the other thing constant.

Now $\nabla (f)=(f_x,f_y,f_z)$ by the above definition can you try to solve the problem ???????

The points on the straight line $\frac{x-1}{2} = \frac{y-2}{-1}= \frac{z}{-2} = k$ $\implies x=1+2k , y=2-k , z=-2k$ So any point on the straight line would be of the form $(x,y,z)= (1,2,0) + (2,-1,-2)k$ Here the direction of the straight line would be $(2,-1,-2)$

Now we have to consider the unit vector in that direction and so the unit vector would be $u= \frac{(2,-1,-2)}{\sqrt{4+1+4}} = (\frac{-2}{3},\frac{-1}{3},\frac{-2}{3})$ Here the absolute value of the directional derivative would be $\nabla \phi . u$ . Would you like to calculate this dot product?

$\nabla \phi = (\phi_{x},\phi_{y},\phi_{z})$ i.e the co-ordinates involving partial derivative of $\phi$ . So, $\nabla \phi = (0, 6y+3z , 3y)|_{(1,-2,1)} = (0,-9,-6)$

Then $\nabla \phi . u= \frac{9}{3} + \frac{6 \times 2}{3} = 3+4 = 7$ (Ans) .

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