Topic 2 of 7

Partial derivatives and gradients

Step 18 · Multivariable calculus topics

Differentiate functions of several variables one variable at a time, gather the results into the gradient, and use it for rates of change in any direction and for tangent planes.

Partial derivatives

For f(x, y), the partial derivative $f_x$ is the ordinary derivative in x with y held fixed: the slope of the surface as you walk in the x direction. $f_y$ works the same way in y.

f_x = \frac{\partial f}{\partial x} \qquad f_y = \frac{\partial f}{\partial y}

Example: A polynomial

Find $f_x$ and $f_y$ for f(x, y) = x³y² + 4xy − y.

For $f_x$ , treat y as a constant: 3x²y² + 4y.
For $f_y$ , treat x as a constant: 2x³y + 4x − 1.

Answer $f_x = 3x^{2}y^{2} + 4y$ and $f_y = 2x^{3}y + 4x - 1$

Example: With the chain rule

Find $f_x$ for $f(x, y) = e^{xy}$ .

Treat y as a constant: the inside, xy, has x-derivative y.

Answer $f_x = y\,e^{xy}$

The gradient

The gradient gathers the partial derivatives into a vector, $\nabla f = \langle f_x, f_y \rangle$ . It points in the direction in which f increases fastest, its length is that fastest rate, and it crosses the level curves at right angles.

Level curves of f = x² + y² are circles. The gradient, drawn as short segments, points straight outward across the circles, toward higher values.

Example: Fastest increase

f(x, y) = x² + 3y. In which direction does f increase fastest at (1, 2), and how fast?

∇f = ⟨2x, 3⟩, which is ⟨2, 3⟩ at (1, 2).
Its length is √13.

AnswerIn the direction ⟨2, 3⟩, at a rate of √13, about 3.61

Tangent planes and directional derivatives

The rate of change of f in the direction of a unit vector u is the directional derivative, $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$ . Near a point, the surface z = f(x, y) is close to its tangent plane.

z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

Example: A directional derivative

For f(x, y) = x² + 3y, find the rate of change at (1, 2) in the direction of ⟨3, 4⟩.

The unit vector is ⟨0.6, 0.8⟩.
∇f = ⟨2, 3⟩, so the rate is 1.2 + 2.4.

Answer3.6

Example: A tangent plane

Find the tangent plane to z = x² + y² at (1, 2, 5).

At (1, 2), $f_x = 2x = 2$ and $f_y = 2y = 4$ .
z = 5 + 2(x − 1) + 4(y − 2).

Answerz = 2x + 4y − 5

Example: Linear approximation

Use that plane to estimate f(1.1, 1.9).

Put x = 1.1 and y = 1.9 into z = 2x + 4y − 5.
2.2 + 7.6 − 5 = 4.8.

Answer4.8; the true value is 4.82

Try it yourself

Work each one on paper first, then open the answer to check.

Find $f_x$ for f(x, y) = 5x²y − y³.
Show answer
10xy
Find $f_y$ for f(x, y) = 5x²y − y³.
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5x² − 3y²
Find the gradient of f(x, y) = xy at (2, 3).
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⟨3, 2⟩
What is the largest rate of increase of f(x, y) = 3x + 4y?
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5
For f(x, y) = xy, find the rate of change at (2, 3) in the direction ⟨1, 0⟩.
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3
Find the tangent plane to z = xy at (1, 1, 1).
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z = x + y − 1