Topic 11 / 17intermediate

Gradient, Directional Derivative, Tangent Planes

Compute gradients and directional derivatives. Find tangent planes and normal lines to surfaces. Use linearization and differentials for approximation.

The gradient $\nabla f$ packages the partial derivatives into a vector. It points in the direction of greatest increase of $f$ , with magnitude equal to the maximum rate of change. Tangent planes and linearization extend the idea of a tangent line to surfaces.

The gradient

\nabla f (x, y) = ⟨ f_{x}, f_{y} ⟩

in 2D,

\nabla f (x, y, z) = ⟨ f_{x}, f_{y}, f_{z} ⟩

in 3D. Properties: it is normal to level curves (in 2D) or level surfaces (in 3D); points in the direction of greatest increase; magnitude

∣\nabla f ∣

equals the max rate of increase.

Key takeaways

Greatest increase: direction $\nabla f$ .
Greatest decrease: direction $- \nabla f$ .
Zero rate of change directions: perpendicular to $\nabla f$ .

Directional derivative

D_{u} f = \nabla f \cdot u

where

u

is a **unit vector**. If you're given a non-unit direction

v

, divide by

∣ v ∣

first. Geometrically: rate of change of

f

along the direction

u

Strategy tips

Always normalize the direction vector.
$D_{u} f$ is signed: positive means $f$ increases in that direction.

Tangent plane to a surface $z = f(x, y)$

z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})

. This is the linearization

L (x, y)

used to approximate

f

near

(x_{0}, y_{0})

Tangent plane to a level surface $F(x, y, z) = k$

Normal direction is

\nabla F

. So the tangent plane satisfies

\nabla F (P_{0}) \cdot ((x, y, z) - P_{0}) = 0

. The normal line is

r (t) = P_{0} + t \nabla F (P_{0})

Differentials

df = f_{x} d x + f_{y} d y

. Use to estimate

Δ f \approx df

for small changes.

Worked examples

Example 1

Find the directional derivative of

f (x, y, z) = x^{2} y + x 1 + z

(1, 2, 3)

in direction

v = ⟨ 2, 1, - 2 ⟩

1
Gradient.
$\nabla f = ⟨ 2 x y + 1 + z, x^{2}, \frac{x}{2 1 + z} ⟩$
2
At $(1, 2, 3)$ .
$\nabla f = ⟨ 4 + 2, 1, \frac{1}{4} ⟩ = ⟨ 6, 1, 1/4 ⟩$
3
Unit vector $\overset{v}{^}$ .
$∣ v ∣ = 3, \hat{v} = \frac{1}{3} ⟨ 2, 1, - 2 ⟩$
4
Dot product.
$D_{\overset{v}{^}} f = \frac{1}{3} (12 + 1 - 0.5) = \frac{12.5}{3} = \frac{25}{6}$

Answer

D_{\overset{v}{^}} f = 25/6

Example 2

Find the tangent plane to

z = e^{x} cos y

(0, 0, 1)

1
Partials.
$f_{x} = e^{x} cos y, f_{y} = - e^{x} sin y$
2
At $(0, 0)$ .
$f_{x} (0, 0) = 1, f_{y} (0, 0) = 0$
3
Plug into formula.
$z = 1 + 1 \cdot (x - 0) + 0 \cdot (y - 0) = 1 + x$

Answer

z = 1 + x

Example 3

Find the tangent plane to the level surface

x^{2} + 2 y^{2} - 3 z^{2} = 3

(2, - 1, 1)

1
$F (x, y, z) = x^{2} + 2 y^{2} - 3 z^{2} - 3$ . Gradient.
$\nabla F = ⟨ 2 x, 4 y, - 6 z ⟩$
2
At $(2, - 1, 1)$ .
$\nabla F = ⟨ 4, - 4, - 6 ⟩$
3
Tangent plane.
$4 (x - 2) - 4 (y + 1) - 6 (z - 1) = 0, or 2 x - 2 y - 3 z = 5$

Answer

4 (x - 2) - 4 (y + 1) - 6 (z - 1) = 0

, equivalently

2 x - 2 y - 3 z = 5

Interactive visualizations

Loading visualization…

point x₀0.50

point y₀0.50

Surface: f(x, y) = x² + y²/2

z \approx 0.375 + 1.000 (x - 0.50) + 0.500 (y - 0.50)

Drag the red point: the tangent plane (red translucent) updates to match the surface's slope at that point.

Loading visualization…

Contour map of

f = x^{2} + y^{2}

with the gradient

\nabla f (1, 1) = ⟨ 2, 2 ⟩

. The gradient is **normal** to the level curve through the point.

Formulas in this topic

Gradient (3D)

\nabla f = ⟨ f_{x}, f_{y}, f_{z} ⟩

Partial derivatives packaged as a vector.

Directional derivative

D_{u} f = \nabla f \cdot u

u must be a unit vector.

Max rate of change

max D_{u} f = ∣\nabla f ∣

Achieved when u is along the gradient.

Tangent plane (z = f)

z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})

Linearization of z = f(x,y).

Tangent plane (level)

\nabla F (P_{0}) \cdot (r - r_{0}) = 0

Tangent plane to F(x,y,z) = k.

Normal line

r (t) = r_{0} + t \nabla F (r_{0})

Line perpendicular to a level surface at a point.

Differential

df = f_{x} d x + f_{y} d y

First-order change estimate.

The gradient

Directional derivative

Tangent plane to a surface $z = f(x, y)$

Tangent plane to a level surface $F(x, y, z) = k$

Differentials

Worked examples

Interactive visualizations

Formulas in this topic

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