CalculusConcept reference

Partial derivatives

A partial derivative measures a scalar function’s rate in one independent coordinate while all other coordinates are held constant.

2 min read1 figureSources & further reading

On this page 9 sections

Overview
Understand the idea
For f(x,y)=x^2+3y, choosing y instead changes the calculation
A partial derivative can itself depend on several coordinates
A closer look
Key takeaway
Further questions
Sources & further reading
Concept connections

01Understand the idea#

A partial derivative measures how a several-input function responds when one input changes and all the others stay fixed. For a model with many weights, it isolates the local effect of one weight on the loss. The variable you differentiate with respect to determines what counts as constant.

A partial derivative changes one input coordinate. For $f(x,y)=x^2+3y$ , the highlighted path varies $x$ and keeps $y=2$ .

Figure 1Input plane for f(x,y)=x squared+3y. The horizontal highlighted path keeps y=2 while x varies from -2 to 2. At fixed y=2 the scalar function becomes x squared+6. The axes use equal 26 pixels per data unit; the path is in input space, not a graph of output values.

For $f(x,y)=x^2+3y$ , treat $y$ as a constant when differentiating in $x$ . The term $3y$ has derivative zero.

f_x=\frac{\partial f}{\partial x}=2x

For $f(x,y)=x^2y+3y$ at $(2,4)$ , hold $y=4$ when finding the rate in $x$ . Symbolically, $f_x=2xy$ , so $f_x(2,4)=16$ . For the rate in $y$ , hold $x$ fixed: $f_y=x^2+3$ , giving 7. The same function has different rates in different input directions.

Check your reasoning

$f(x,y)=4xy+y^2$ . Find $f_x$ .

A $4x+2y$
B $4y$
C $4y+2y$

Show answer and explanation

$4y$

Hold $y$ fixed: $f_x=4y$ .

02For f(x,y)=x^2+3y, choosing y instead changes the calculation#

For $f(x,y)=x^2+3y$ , choosing $y$ instead changes the calculation. Now $x^2$ is constant, and $3y$ has derivative 3.

f_y=3

Keep the inner derivative in the chain rule. For $g(x,y)=(x+3y)^2$ , hold $x$ fixed when finding $g_y$ . The inner derivative is 3.

g_y=6(x+3y)

Check your reasoning

$f(x,y)=(2x+y)^2$ . A learner writes $f_x=2(2x+y)$ . Repair it.

A $2(2x+y)$
B $2(2x+y)(2+1)$
C $4(2x+y)$

Show answer and explanation

$4(2x+y)$

The inner derivative in $x$ is 2.

03A partial derivative can itself depend on several coordinates#

A partial derivative can itself depend on several coordinates. For $f(x,y)=x^2y$ , first find $f_x=2xy$ , then evaluate at $(2,3)$ .

f_x(2,3)=12

Check your reasoning

Cost $C(a,b)=a^2+ab$ . At $(a,b)=(2,3)$ , find $C_a$ with $b$ fixed.

Show answer and explanation

$C_a=2a+b=7$ .

04A closer look#

A positive partial derivative means a small increase in that input raises the output locally, with the other inputs fixed. It is a local mathematical sensitivity, not by itself a causal claim about changing a measured feature in the real world.

Key takeaway

Differentiate one coordinate; keep the others fixed.

Compute a partial derivative with other coordinates held fixed.

Further questions

Does a fixed coordinate disappear from the answer?

Not necessarily. In xy, the x partial is y because y stays as the constant coefficient. A term containing only fixed coordinates does have derivative zero.

Sources & further reading

[1]
OpenStax Calculus Volume 3, 4.3 Partial Derivatives ↗OpenStax · Book

01Understand the idea#

02For f(x,y)=x^2+3y, choosing y instead changes the calculation#

03A partial derivative can itself depend on several coordinates#

04A closer look#

Differentiate one coordinate; keep the others fixed.

Further questions

Sources & further reading

Reference this concept