What is Directional derivative: Definition and 102 Discussions

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
The directional derivative is a special case of the Gateaux derivative.

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  1. D

    Directional Derivative Question

    Homework Statement There exists no function f: ℝn \rightarrow ℝ so that for some point a \in ℝn, we have Dvf(a) > 0 for all nonzero vectors v \in ℝn. 2. The attempt at a solution No quite sure where to go with this one, any help would be great!
  2. P

    Directional derivative of a surface

    Homework Statement What is the directional derivative of the function z = x3 - y at the point (1, 2, -1) and in the direction of a vector (1,1,1)? Homework Equations The Attempt at a Solution If f(x,y) = x3 - y, then ∇f = (3x2, -1) which equals (3, -1) at the given point. Now I...
  3. G

    Confused with directional derivative definition of vectors

    Hi, been reading about GR and am quite confused about the new definition of vectors. My main problem with this is that the text uses partial derivatives as the vector basis, I understand this is related to directional derivatives but cannot see the direct mathematical relation. Secondly, how...
  4. S

    MHB Directional derivative

    Let f(x,y) = (x-y)/(x+y). find the directions u and the values of $D_{u}f $ (-1/2 , 3/2) for which $D_{u}f $ (-1/2 , 3/2) is largest, and is smallest. How do i go about that? I did it for when $D_{u}f $ (-1/2 , 3/2) = 1 and got $D_{u}f $ (-1/2 , 3/2) = 1 and got u=j and -i. This was after i...
  5. 4

    Directional Derivative for f(x)

    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 ?
  6. 1

    Gradient vs. Directional Derivative

    On a quiz, a true/false statement was given along the lines of: "The gradient is a specific example of a directional derivative." I marked "true" and got it wrong. I see why, I think, since the gradient is an actual "guide," a vector, towards the max rate of change, while the directional...
  7. M

    Directional derivative tangent to a curve

    Homework Statement Given xz^2-yz+cos(xy)=2 which defines z implicitly in terms of x and y, find the directional derivative of z in the direction of the tangent to the curve y=x^2+2x-1 at the point (0,1) in the direction of decreasing x Homework Equations The Attempt at a Solution...
  8. D

    Calculate the directional derivative

    Homework Statement Calculate the directional derivative, direction, and rate of maximum increase in the direction of v at the given point. f(x,y) = x^2 + y^3 v = <4,3> P = (1,2) Homework Equations I know how to calculate directional derivative but I don't know how to calculate rate of maximum...
  9. Useful nucleus

    Directional Derivative of a discontinuous function

    I was reviewing basic calculus of of functions of several variables. It struck me that in one of the exercises, it was required to show that a given function has directional derivative for all directions at the origin (0,0) even though it is not continuous at (0,0). Without getting into the...
  10. T

    Directional derivative and the gradient - confused.

    Hello, thanks for reading! I am slightly confused. According to the definition of the directional derivative, calculated at the point x in the direction y, f'(x;y) = lim\frac{f(\vec{x} + h\vec{y})-f(\vec{x})}{h}, h-->0 According to this definition, the directional derivative seems to not...
  11. estro

    Directional derivative and non-appropriate solution?

    Directional derivative and "non-appropriate" solution? Homework Statement Given following normal vectors u_1,__2,...,u_n which are basis for R^N Prove that there are n scalars which satisfy: f_x(a)=\sum_{i=1}^na_if_{u_i}(a), given that f is differentiable at a\in R^n Homework Equations I know...
  12. P

    Finding the Directional Derivative of a Multivariable Function

    find the directional derivative of z=2x^2-y^3 at (1,1) is it just <4,-3>
  13. K

    Directional Derivative - Right Steps?

    1. Find the directional derivative of the function f(x,y) = e^{xy} at the point (-2,0) in the direction of the unit vector that makes an angle of \pi/3 with the positive x-axis The vector to make into a unit vector is (\sqrt{3},1) and e^{xy} differentiated with respect to x and y is both...
  14. B

    Solution to pde with directional derivative first order

    Homework Statement trying to solve v.\nabla_x u + \sigma(x) u = 0 (x,v) \in \Gamma_- \Gamma_- = \left\{(x,v) \in X x V, st. -v.\nu(x) > 0\right\} \nu(x) = outgoing normal vector to X v = velocity u = density g(x) = Incoming boundary conditions The Attempt at a...
  15. jegues

    Finding the Directional Derivative of a Function at a Given Point on a Curve

    Homework Statement Find the rate of change of the function, f(x,y,z) = cos(\pi x y) + xln(z^{2}+1), with respect to length s along the curve, y=-3x, z=x^{2}-y^{2}+9, directed so that x increases, at the point (-1,3,1). Homework Equations The Attempt at a Solution See...
  16. H

    Why isn't my directional derivative coming out right?

    Homework Statement Find the directional derivative of f(x, y) = (xy)1/2 at P(6, 6) in the direction of Q(2, 9) Homework Equations D_uf(x,y) = \nablaf*u : dot product The Attempt at a Solution \nablaf = <y1/2/(2x1/2),x1/2/(2y1/2)> vector PQ = Q-P = <2-6,9-6> = <-4,3> unit...
  17. L

    Rate of change of temperature at (1,-1,1) towards (2,4,-4)

    Homework Statement The temperature at a point (x,y,z) is given by ,T(x,y,z)=200e^((-x^2-y^2)/(4-z^2)/9) where is measured in degrees Celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector. Find...
  18. B

    Directional Derivative Question

    Here's the problem: Let f(x,y) = sqrt(|xy|) if x >= 0 OR -sqrt(|xy|) if x < 0. Show that all directional derivatives in the direction of h exist, but f is not differentiable at (0,0). Here's what I've done: Basically all I need to show is the directional derivative is not linear. Since if...
  19. K

    Solving Directional Derivative Problems: A Comprehensive Guide

    nvmnd - solved
  20. C

    Directional derivative and gradient concepts

    Homework Statement A series of true/false questions. I guess I don't understand the concepts of this very well: 1. If you know the directional derivative of f(x,y) in two different directions at a point P, we can find the derivative with respect to the x and y axes and thus we can...
  21. G

    Directional Derivative of a Function in a Given Direction

    Homework Statement Find the Directional derivative of \varphi = x2+siny-xz in the direction of i+2j-2k at the point (1, \pi/2 , -3) The Attempt at a Solution u = i+2j-2k \left|u\left| = \sqrt{1^2+2^2+(-2)^2} = 3 \Rightarrow u\hat{} = 1/3i+2/3j-2/3k \nabla\varphi =...
  22. P

    Directional Derivative and Approach Path

    Homework Statement I have a problem that says: Find the directional derivative of C(x, y) in the radial direction at any surface point (x, y). It then says: Find the shark’s approach path from any point (x0, y0) at the sea surface. I found the directional derivative in the radial...
  23. W

    Calculating the directional derivative of a function of two variables

    Homework Statement Consider the function: z=f(x,y)= log(x^2 + y^2) (x,y)=/=(0,0) Calculate the directional derivative of f(x,y) at (x,y)=(1,1) in the direction of the vector (1,2) The attempt at a solution When i tried to work out the unit vector from the point (1,1) to...
  24. V

    Gradient exists but directional derivative does not?

    First of all, I don't have a concrete example for this, but I hope it's not too hard to understand what I'm trying to get at. For a multivariable function of, say, 2 variables x and y, the gradient at a point only depends on the existence of partial x and partial y, right? In other words, if...
  25. U

    Find a vector such that directional derivative is zero

    Homework Statement Let f (x, y) = e^x^2 + 3e^y . At the point (0, 1) find: (a) a vector u such that the directional derivative D_u f is maximum and write down this maximum value, (b) a vector v such that D_v f = 0 Homework Equations grad f / directional derivative formulaThe Attempt at a...
  26. S

    Directional Derivative of F: Same for V & 2V?

    A very basic question: Is the directional derivative of a scalar function F at some point 'a' in the direction V the same as the directional derivative of F at 'a' in the direction 2V? Going by http://en.wikipedia.org/wiki/Directional_derivative#Definition" definition of a directional...
  27. E

    Directional derivative with angle

    Homework Statement Find the directional derivative of the function given below as you arrive at (0,1,3) from the direction of (1,1,0). Give an exact answer. f(x,y,z) = sin(5 x) + ln(y^2+1) + z^3 Homework Equations The Attempt at a Solution I know how to solve this if it were a 2...
  28. M

    Maximizing Function Increase: Understanding Directional Derivatives

    I have a function of 2 variables. I know it increase most rapidly in the direction of the gradient, but how about in wht direction is it not increasing? I am thinking that the gradient (dot product)(direction in which it is not increasing) = 0 Any hints?
  29. Z

    Directional derivative without reparametrization

    Hi, Everyone, I have the following problem. Given function is g (s, t) = s ^ 2 * exp (t). Find directional derivative of the function at the point (2,0) in direction of vector v = i + j. The book says to find del g with respect to s and t and dot it with normalized v. But I disagree because...
  30. F

    Directional Derivative of a Multivariable Function

    Homework Statement Find the directional derivative of f at P in the direction of a. f(x,y)=y2lnx; P(1,4); a= -3i + 3j Homework Equations The Attempt at a Solution I first normalized the vector and got an = -3/(18)^1/2 + 3/(18)^1/2 Then i took the partial derivative and multiplied...
  31. C

    Determining the Directional Derivative

    [SOLVED]Determining the Directional Derivative Homework Statement Find the following directional derivative of the following function in the direction of the given vector v f(x,y) = x^5 y^3 + 8 \\\ v = (-4,3) Homework Equations \nabla_u f\left(x,y,z\right) =...
  32. A

    Find the Directional derivative of the function u

    Homework Statement Hey. I need to find the Directional derivative of the function u, in the direction l, in the point m (all in the pic). I'm not sure I did it right, can you please give me the ok? Homework Equations The Attempt at a Solution
  33. M

    How do I find the directional derivative?

    Find the directional derivative of f(x,y)= ln sqrt(x^2+y^2) at (x,y) does not = (0,0) toward the origin I found the gradient: grad f(x,y)=1/(x^2+y^2)*(xi+yj) but I have no idea what to do after that.
  34. G

    Specific slope for directional derivative?

    Is there a direction where the rate of change is 18, at the point (-1,2) on the function f(x,y) = (x^2)(y^3)+xy? So I found the gradient of this function, picked a random direction vector u = ( a / (a^2+b^2)^(1/2) , b / (a^2+b^2)^(1/2) ) and took the dot product, and set it to 18... however...
  35. A

    Directional Derivative for Linear Maps

    Hi: Can someone point how to approach this problem- we had 5 problems on directional derivatives and I solved 4. I understand the concept but in this question I don't know where to begin Problem Statement Assume that f:R^{n} -> R^{m} is a linear map, with matrix A with respect to the...
  36. M

    Directional derivative formula

    Homework Statement (Q) The derivative of f(x,y) at Po(1,2) in the direction i + j is 2sqrt(2) and in the direction of -2j is -3. What is the derivative of f in the direction of -i - 2j? Give reasons for your answers. Homework Equations The directional derivative is given by the...
  37. A

    Vector Calc - Directional Derivative Question

    Homework Statement Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point (x,y) is given by M(x,y) = 4x^2 + y^2 + 5000. If Igor is at the point (8,6), describe the curve along which he should travel if he wishes to reduce the field...
  38. K

    Directional derivative and gradient vector

    Homework Statement Find the directional derivative of f=sqrt(xyz) at P(2,-1,-2) in the direction of v=i+2j-2k The Attempt at a Solution I calculate the gradient vector and obtain grad(f) at P= <1/2, 1, 1/2> Then I find the unit vector of v, which is <1/3, 2/3, -2/3> The...
  39. C

    Directional derivative of vector valued functions?

    Just to confirm, is the directional derivative of a vector valued function calculated as Lu ? where L is the Frechet derivative , and u is the unit vector in the direction. There seem to be a lot of sources for a real valued function's directional derivative, but very little on vector valued...
  40. C

    Directional derivative?

    Im doing the following question: calculate the directional derivative of the function f(x,y,z) = z/(2x + y) at the point (0,1,1) in the direction d = 2i - 2j - k could someone please check my answer is correct as i calculated -3i -6k Also how do i find the unit vector in the direction...
  41. C

    Directional derivative without geometry

    I have checked not few books in regard to directional derivatives. Practically all of them use vectors in the derivative definition. I would like to read a good explanation of this concept but keeping out the geometrical interpretations. Suppose we got a function of two variables. I...
  42. Y

    Directional derivative: radial direction

    (hmm. wasnt to sure about where to post this) im given an equation to a 3d surface and asked to find the gradient at a certain point, in the radial direction. my question is, what is the radial direction? [ the equation is f(x,y) = 3*(x^2)*y + 2*y if its needed ]
  43. S

    How Do You Solve This Directional Derivative Problem?

    Hello, I am confused in as how to do this question :S... any help, pointers etc. would be great. http://img87.imageshack.us/img87/637/picture4nd7.png Thanks sarah :)
  44. G

    Directional Derivative and unit vectors

    What happens if a unit vector is not used in calculating the directional derivative. From when I worked it out the directional derivative is multiplied by a scalar if a unit vector is not used. So I gather that the directional derivative must be calculated by a unit vector. Because it is still...
  45. T

    Finding the Directional derivative

    Find the directional derivative of f (x, y) = 5−x2−2y2 at P (-1, -1). anybody pls help as i don't know how to find the direction from this ques so , i can't find the directional derivative pls help thanx
  46. G

    Directional derivative

    Suppose I have a path \vec{r}(x,y) and some vector \vec{a}(x,y). Question is: how do I find the tangential and perpendicular component of a along the path r at a given point? For tangential component, I'd just take the projection of a on r with dot product (I guess this's correct). But what...
  47. Pengwuino

    Minimizing a directional derivative

    I know that you determine maximum value of a directional derivative at a point by finding... |\nabla f(a_0 ,b_0 )| But how do you find the minimum value? I'm also kinda wondering exactly what a gradient is. It seems like if you have the gradient equation and a point... all you are getting...
  48. T

    Calculating the Directional Derivative of T at (1,1,2)

    T(x,y,z) = x^2 - y^2 + z^2 + xz^2 3x^2 - y^2 - z = 0 2x^2 + 2y^2 - z^2 = 0 \bigtriangledown T (x,y,z) = (2x + z^2, -2y, 2z(1+x)) \bigtriangledown T (1,1,2) = 2(3, -1, 4) D_{\overrightarrow{v}}(1,1,2) = \bigtriangledown T(1,1,2).\overrightarrow{v} = 2(3v_1 - v_2 +...
  49. M

    Directional Derivative Problem

    Hello! I have a question about a problem I am trying to work out. The question asks to find the directional derivative of f(x,y,z)=x^2+yz at the point (1,-3,2) in the direction of the path r(t)=t^2i + 3tj+(1-t^2)k. Ok, first do I find the derivative of r(t) to get 3j +t(2i-2k) and then use...
  50. M

    Calc 3 directional derivative question

    The question asks: Find the directional derivative of f (x, y, z) = z ln (x/y) at (1, 1, 2) toward the point (2, 2, 1). What I did was find the distance between the two points to be the directional vector (i+j-k) and then I took the norm of the direction vector. so my unit vector =...
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