← Math 21a: Multivariable Calculus

Unit 3: Partial Derivatives and the Gradient

Slicing a surface to turn slopes into single-variable calculus, then bundling every slope at a point into one arrow: the gradient.

13 topics · 45 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

3.1 Limits in Several Variables

  1. How it worksHow does testing several different paths toward a point expose a two-variable limit that does not exist?
  2. Common mistakeWhy isn't a two-variable limit guaranteed to exist just because it exists along the x-axis and along the y-axis?
  3. Worked exampleFind the limit of (x2 - y2)/(x - y) as (x, y) approaches (2, 2).
Your turn1 of 2
Find the limit of (x2 - y2)/(x - y) as (x, y) approaches (3, 3).

3.2 Continuity and the Path Test for Limits

  1. How it worksHow does comparing the paths y = 0 and y = x expose a limit that depends on the direction of approach?
  2. Worked exampleShow that the limit of xy/(x2 + y2) as (x, y) approaches (0, 0) does not exist by comparing the paths y = 0 and y = x.
  3. Going furtherWhat happens to the path test when you approach along a curved path like y = x2 instead of a straight line?
Your turn1 of 2
Along the straight path y = k·x, the ratio xy/(x2+y2) simplifies to a constant that depends on k. Write that constant as a function of k.

3.3 Partial Derivatives

  1. How it worksHow does holding y fixed and differentiating with respect to x alone produce the partial derivative fx?
  2. Common mistakeWhy doesn't fx measure the surface's steepness in every direction, only in the one direction where y stays fixed?
  3. Worked exampleFind the partial derivatives fx and fy for f(x, y) = x2 y + sin(y).
Your turn1 of 2
Find fx for f(x, y) = 4x2·y + sin(y), as a function of x and y.

3.4 Higher-Order Partial Derivatives and Clairaut's Theorem

  1. How it worksHow do you compute the mixed partial derivatives f_xy and f_yx from a function of two variables?
  2. Common mistakeWhy isn't it true for every possible function that f_xy always equals f_yx?
  3. Worked exampleFind f_xy and f_yx for f(x, y) = x3 y2 + exy, and confirm they agree.
  4. Going furtherWhat happens to the equality of f_xy and f_yx at a point where those mixed partials are not continuous?
Your turn1 of 2
Find f_xy for f(x, y) = x3·y2 + e3·x·y, as a function of x and y.

3.5 The Chain Rule in Several Variables

  1. How it worksHow does the multivariable chain rule add up the contributions from every intermediate variable's own rate of change?
  2. Worked exampleFind dz/dt for z = x2 + y2, where x = cos(t) and y = sin(t).
  3. Going furtherWhat happens to the chain rule formula when z depends on x and y, which each depend on two new variables s and t instead of one?
Your turn1 of 2
Let z = x2·y, x = 3·t, y = 3·t2. Find dz/dt using the chain rule, as a function of t.

3.6 Implicit Differentiation in Several Variables

  1. How it worksHow do the partial derivatives of F(x, y) combine into the formula dy/dx = -Fx / Fy for an implicit curve?
  2. Common mistakeWhy doesn't implicit differentiation require solving the equation for y before you can differentiate it?
  3. Worked exampleFind dy/dx at the point (1, 1) on the curve x3 + y3 = 2 using implicit differentiation.
Your turn1 of 2
The curve x3 + 4·y3 = 13 defines y implicitly as a function of x. Find dy/dx, as a function of x and y.

3.7 Tangent Planes to a Surface

  1. How it worksHow do the two partial derivatives at a point combine into the normal vector that defines the tangent plane?
  2. Worked exampleFind the equation of the tangent plane to z = x2 + y2 at the point (1, 2, 5).
  3. Going furtherWhat happens to the tangent plane at a point where both partial derivatives equal zero?
Your turn1 of 2
Find the tangent plane to z = x2 + 4·y2 at the point (1, 3, z0), where z0 = x02 + 4·y02. Write that tangent plane's z as a function of x and y.

3.8 Linearization and the Differential

  1. How it worksHow does the tangent plane's equation double as a linear approximation formula near the point of tangency?
  2. Worked exampleFind the linearization-based estimate of the square root of (4.022 + 3.012) near the point (4, 3).
  3. Common mistakeWhy doesn't the linear approximation stay accurate far away from the point where it was built?
Your turn1 of 2
Linearize f(x, y) = 4·x2 + 4·y2 at the point (3, -2). Write the linearization L(x, y) as a function of x and y.

3.9 The Gradient Vector

  1. How it worksHow do the partial derivatives fx and fy stack together into the gradient vector at a point?
  2. Worked exampleFind the gradient of f(x, y) = x2 y + 3y at the point (2, 1).
  3. How it worksHow does the gradient vector at a point always end up perpendicular to the level curve passing through that same point?
  4. Common mistakeWhy doesn't the gradient point toward the function's maximum value, only toward the steepest way uphill from where you stand?
  5. Going furtherWhat happens to the gradient vector's length at a point where the level curves bunch tightly together?
Your turn1 of 2
Find the y-component of the gradient of f(x, y) = 4·x2·y + 4·y, as a function of x.

3.10 Directional Derivatives

  1. How it worksHow does taking the dot product of the gradient with a unit direction vector produce the directional derivative?
  2. Common mistakeWhy does the directional derivative formula require the direction vector to be a unit vector before you use it?
  3. Worked exampleFind the directional derivative of f(x, y) = x2 + xy at (1, 2) in the direction of the vector <3, 4>.
  4. Going furtherWhat happens to the directional derivative as the chosen direction rotates all the way around to point opposite the gradient?
Your turn1 of 2
Find the directional derivative of f(x, y) = 4·x2·y in the direction (4, 6), as a function of x and y.

3.11 The Gradient as the Direction of Steepest Ascent

  1. How it worksHow does maximizing the dot product of the gradient with a unit vector prove the gradient points in the steepest-ascent direction?
  2. Worked exampleFind the direction of steepest ascent for f(x, y) = x2 + y2 at the point (3, 4).
  3. Ties togetherWhy is the gradient perpendicular to a level curve and also the normal vector of the tangent plane?
Your turn1 of 2
For f(x, y) = 4·x2 + 4·y2, find the x-component of the unit vector pointing in the direction of steepest ascent, as a function of x and y.

3.12 The Gradient in Three Variables and the Normal to a Level Surface

  1. How it worksHow does the gradient of F(x, y, z) at a point become the normal vector to the level surface F = c passing through that point?
  2. Worked exampleFind the normal vector to the level surface x2 + y2 + z2 = 9 at the point (0, 0, 3), using the gradient of F(x, y, z) = x2 + y2 + z2.
  3. Common mistakeWhy isn't the gradient of a three-variable function itself a vector lying tangent to the level surface at that point?
  4. Going furtherWhat happens to the gradient-as-normal-vector idea at a sharp point of a level surface, like the tip of a cone?
Your turn1 of 2
For F(x, y, z) = 4·x2·y + 4·y·z, find the x-component of grad F (the normal vector to the level surface F = k), as a function of x and y.

3.13 Tangent Lines to the Intersection of Two Surfaces

  1. How it worksHow does taking the cross product of the two surfaces' gradient vectors at a point produce the tangent direction to their curve of intersection?
  2. Worked exampleFind the tangent line direction to the curve of intersection of x2 + y2 + z2 = 3 and x + y + z = 3 at the point (1, 1, 1).
  3. Common mistakeWhy isn't the tangent line to a curve of intersection simply the average of the two surfaces' individual normal vectors?
  4. Ties togetherHow does taking the cross product of two gradient vectors reuse the same cross-product idea from vector geometry?
Your turn1 of 2
Surfaces F(x, y, z) = 4·x2 + y2 + z2 = k and G(x, y, z) = x + y + z = m intersect in a curve. Find the z-component of grad F x grad G (the curve's tangent direction), as a function of x and y.