Textbook
The function $f:\mathbb R^d\to\mathbb R$ is defined in an open neighborhood of $x$. The partial derivative in coordinate $x_i$ is $\frac{\partial f}{\partial x_i}(x)=\lim_{h\to0}\frac{f(x+he_i)-f(x)}{h}$ when this finite limit exists. Here $e_i$ has a 1 in position $i$ and zeros elsewhere.
Only one coordinate changes in this limit. Every other coordinate stays fixed at its value in the chosen point. A partial derivative is therefore an ordinary derivative of a one-dimensional slice.
For $f(x_1,x_2)=x_1^2+3x_2^2$, the formulas are $\partial f/\partial x_1=2x_1$ and $\partial f/\partial x_2=6x_2$. At $(1,2)$ they equal 2 and 12. For the first derivative, $x_2$ stays fixed, so $3x_2^2$ cancels in the difference.
The existence of all partial derivatives at one point does not guarantee a linear approximation for simultaneous coordinate changes. Continuous partial derivatives in a neighborhood provide a convenient sufficient condition for differentiability. The symbol $\partial$ identifies the chosen coordinate but guarantees none of these assumptions.
For the illustrative function at $(1,2)$, the first-coordinate difference quotient is $2+h$ and the second-coordinate quotient is $12+3h$ for $h\ne0$. The extra terms disappear as $h$ approaches zero. A finite quotient differs from the limiting derivative: it gives average change along the chosen slice segment, while the derivative gives the local coefficient at the initial point.
Definitions and notation: Basic Analysis I; Mathematics for Machine Learning.
Related concepts: Derivatives: local rates of change.
In plain language
A partial derivative resembles testing one control while every other control stays fixed. Turning two knobs at once mixes their effects, and changing the starting point creates a different experiment. For the illustrative function at $(1,2)$, the values 2 and 12 belong to that point and the chosen coordinate units. A permanent “chief control” badge would be premature: the result describes the local response to changing one coordinate.
Example
For the constructed function $x_1^2+3x_2^2$, the initial point is $(1,2)$ and each coordinate increases separately by 0.1.
Partial derivatives: changing one coordinate
Initial point
The initial value is 1 + 12 = 13.
Step 1 of 4: Initial point
Read all steps
Initial point
The initial value is 1 + 12 = 13.
- Coordinates
- 12
- Function
- 13current
First-coordinate change
The second coordinate stays 2; the value becomes 13.21.
- Coordinates
- 1.1current2
- Difference quotient
- 2.1current
Second-coordinate change
The first coordinate returns to 1 and only the second changes to 2.1.
- Coordinates
- 12.1current
- Function; quotient
- 14.23current12.3current
Smaller step
At h = 0.01 the quotients are closer to their partial derivatives.
- First; second coordinate
- 2.01included12.03included
- Limits
- 2included12included
At $h=0.1$, the quotients are 2.1 and 12.3; at $h=0.01$, they are 2.01 and 12.03. Their differences from the derivatives shrink tenfold with the step, as the exact formulas above show. The second experiment starts again at $(1,2)$, not $(1.1,2)$. Thus the two slices share one point; mixing their movements into a single path would change the question being investigated.
A partial derivative requires identifying the changing coordinate and everything held fixed.