Partial Derivative

Definition

A partial derivative measures how a function changes with respect to one input while the other inputs are held fixed.

Example function:

f(x, y) = x + 2y

This function has more than one input.

We can ask:

How does f change when x changes?
How does f change when y changes?

Partial derivative = one-input-at-a-time derivative.

---

Why Partial Derivatives Are Needed

Many useful functions depend on more than one value.

Example:

score = w1*x1 + w2*x2 + b

The score depends on:

w1
w2
b
x1
x2

If one value changes, the score may change.

Partial derivatives let us measure each effect separately.

Multi-input functions need directional sensitivity for each input.

---

Holding Other Inputs Fixed

For:

f(x, y) = x + 2y

If we change x while holding y fixed:

partial with respect to x = 1

If we change y while holding x fixed:

partial with respect to y = 2

This means y has twice the direct effect in this function.

Holding other inputs fixed isolates one source of change.

---

Not The Whole Story At Once

A partial derivative does not describe every possible change at once.

It answers a focused question:

What happens if this one input changes and the others stay the same?

If multiple inputs change together, their combined effect requires combining information.

That leads to gradients.

A partial derivative is one component of multi-input change.

---

Partial Derivatives And Parameters

In learning systems, parameters can be treated as inputs to a loss function.

Example:

loss = function(weight1, weight2, bias)

Then we can ask:

How does loss change if weight1 changes?
How does loss change if weight2 changes?
How does loss change if bias changes?

Each question is a partial derivative.

Partial derivatives measure parameter sensitivity one parameter at a time.

---

Why This Is Still Local

A partial derivative is measured at the current point.

Example:

current x
current y

The answer describes nearby behavior around that current combination.

It does not automatically describe every region of the function.

Partial derivatives are local measurements in multi-input functions.

---

Final Summary

• A partial derivative measures change with respect to one input.
• Other inputs are held fixed.
• Multi-input functions need partial derivatives.
• Each partial derivative measures one source of sensitivity.
• Gradients collect partial derivatives into one object.