# What is a function, when you consider the partial/total derivative?

• mXSCNT
In summary: All derivatives in calculus are written in terms of these two operations, and there is no need for a separate notation for the total derivative.
mXSCNT
Example, if:
$z=xy$
$y=x$
then:
the partial derivative $\frac{\partial z}{\partial x} = y$, treating $z$ as a function of two variables $z(x, y) = xy$
the total derivative $\frac{dz}{dx} = 2x$, treating $z$ as a function of one variable, $z(x) =z y(x)$.

So there doesn't seem to be a way to define the partial or total derivative based only on one function $z$ defined as a set of ordered pairs plus domain/codomain. So how do you define a function in such a way that you can define the partial and total derivative?

This may not be clear. If you don't see the problem, consider this question:
For $f : \mathbb{R} \times \mathbb{R} -> \mathbb{R}$, what's the total derivative of $f$ with respect to its first parameter, and how does this differ from the partial derivative of $f$ with respect to its first parameter?

How many parameters does the operator "total derivative" take and what are they? There's at least the function and the variable the total derivative is taken with respect to, but those two parameters don't seem enough, without changing the definition of function. How many parameters does the operator "partial derivative" take and what are they?

To handle the distinction between the total derivative and partial derivative, in calculus you also need some idea of the relationships between the variables used in the function definition. These relationships are not encoded in the usual definition of a function.

Last edited:
hi mXSCNT!
mXSCNT said:
To handle the distinction between the total derivative and partial derivative, in calculus you also need some idea of the relationships between the variables used in the function definition.

correct … ∂/∂x depends on what the other variable(s) are, since you have to calculate it while keeping all the other variables constant

so eg ∂/∂x f(x,y,z) = d/dx fy,z(x), where fy,z(x) = f(x,y,z)

(this is not a standard notation)

mXSCNT said:
Example, if:
$z=xy$
$y=x$
then:
the partial derivative $\frac{\partial z}{\partial x} = y$, treating $z$ as a function of two variables $z(x, y) = xy$
the total derivative $\frac{dz}{dx} = 2x$, treating $z$ as a function of one variable, $z(x) =z y(x)$.

So there doesn't seem to be a way to define the partial or total derivative based only on one function $z$ defined as a set of ordered pairs plus domain/codomain. So how do you define a function in such a way that you can define the partial and total derivative?

You don't.

Given a differentiable $f: \mathbb{R}^2 \to \mathbb{R}$ you can define these partial derivatives (I'm avoiding the Liebnitz notation because it tends to obscure matters in rigorous analysis):
$$f_1: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}h \\ f_2: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto \lim_{h \to 0} \frac{f(x,y+h) - f(x,y)}h$$
These respectively are the partial derivatives of $f$ with respect to its first and second arguments.

Given a differentiable $f: \mathbb{R}^2 \to \mathbb{R}$ and a differentiable $g: \mathbb{R} \to \mathbb{R}$, you can then define an entirely new function
$$h: \mathbb{R} \to \mathbb{R}: x \mapsto f(x,g(x))$$
whose derivative can be shown to be
$$h' : x \mapsto f_1(x, g(x)) + f_2(x, g(x))g'(x)$$
Note that $f \neq h$ since the domains are different: the domain of the first is $\mathbb{R}^2$ and the domain of the second is $\mathbb{R}$. In these circumstances, writing
$$\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}$$
is a notational abuse (one of the many to which the Liebnitz notation lends itself).

## 1. What is a function?

A function is a mathematical relationship between two or more variables, where each input (independent variable) corresponds to a unique output (dependent variable).

## 2. How is a function represented?

A function can be represented in several ways, including algebraically (using equations or formulas), graphically (using a graph or plot), or numerically (using a table of values).

## 3. What is a partial derivative?

A partial derivative is the rate of change of a function with respect to one of its variables, while holding all other variables constant. It measures how much the output of a function changes when only one input variable is changed.

## 4. What is a total derivative?

A total derivative is the rate of change of a function with respect to all of its variables. It takes into account the effect of changing all input variables on the output of the function. It is also known as the gradient or the derivative vector.

## 5. Why are partial and total derivatives important?

Partial and total derivatives are important in mathematics and science because they allow us to analyze and understand the behavior of complex functions with multiple variables. They are used in fields such as economics, physics, and engineering to model and predict changes in systems. They also play a crucial role in optimization and calculus.

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