Directional Derivative for f(x)

[4570562]

Homework Statement

Is it possible to take the directional derivative of a function of a single variable?
For example if f(x)=sinx and the unit vector specifying the direction is u=<cos pi/4, sin pi/4>,
then could you say that duf = .5*(sqrt2)*cosx ?

 

[Dick]

I vote no. The question looks meaningless to me. If they had said f(x,y)=sin(x) then I might agree with that.

 

[LCKurtz]

Sure, you can do it in 1D. Say your function is $f(x)=-x^2$ Then $\nabla f = -2x\vec i$. Now suppose, for example, this function represents the temperature in a 1D rod along the x-axis. At $x=2$ we have $\nabla f(2) = -4\vec i$. Notice that the gradient points the direction of maximum increase in temperature ($-\vec i$) and its magnitude gives the rate of change in that direction.

 

[Dick]

Sure, you can do it in 1D. Say your function is $f(x)=-x^2$ Then $\nabla f = -2x\vec i$. Now suppose, for example, this function represents the temperature in a 1D rod along the x-axis. At $x=2$ we have $\nabla f(2) = -4\vec i$. Notice that the gradient points the direction of maximum increase in temperature ($-\vec i$) and its magnitude gives the rate of change in that direction.

True, but what can you do with "the unit vector specifying the direction is u=<cos pi/4, sin pi/4>"?

 

[4570562]

Maybe you can take the gradient of a function of a single variable but just not the directional derivative? It seems to me now that the directional derivative wouldn't make any sense for f(x) because you can't vary x in a "direction". Varying x in the negative or positive direction still gives you the same result for the derivative, I think.

But since duf=gradf . u I think they might both have to go.

 

[HallsofIvy]

There are only two directions in one dimension- "left" and "right" (as opposed to an infinite number of directions in two or more dimensions).

You can think of the two directional derivatives of f(x) at x= a as f'(a) (to the right) and -f'(a) (to the left).

 

[LCKurtz]

True, but what can you do with "the unit vector specifying the direction is u=<cos pi/4, sin pi/4>"?

Nothing, of course. I just answered the question I think he meant to ask.