Value of partial derivative depends on all inputs

Statement

The general expression for the partial derivative of a function with respect to any of the inputs is an expression in terms of all the inputs. For instance, the general expression for $f_{x}(x,y)$ is an expression involving both $x$ and $y$. This is because, even though the $y$-coordinate is kept constant when calculating the partial derivative at a particular point, that constant value need not be the same at all the points.

Example

Mathematical example involving two variables

For instance, consider:

$f(x,y):=x^{2}+y^{2}+xy^{2}$

Then, we have:

$f_{x}(x,y)=2x+y^{2}$

and:

$f_{y}(x,y)=2y+2xy$

Note that each of the expressions involves both the variables $x$ and $y$. In particular, this means that the value of $f_{x}$ at a point depends on both the $x$-coordinate and the $y$-coordinate of the point. Thus, for instance:

$f_{x}(2,3)=2(2)+3^{2}=4+9=13$

$f_{x}(2,4)=2(2)+4^{2}=4+16=20$

Despite the same $x$-value of 2 in both cases, the $f_{x}$-values are different because of differences in the input $y$-values.

Similarly, consider:

$f_{y}(1,4)=2(4)+2(1)(4)=8+8=16$

$f_{y}(2,4)=2(4)+2(2)(4)=8+16=24$

Despite the same $y$-value of 4 in both cases, the $f_{y}$-values are different because of differences in the input $x$-values.

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Real-world example

The real-world example mentioned here uses something that's a relative derivative between logarithms of quantities, but the idea is the same. In standard microeconomic theory, the quantity demanded for a particular good is considered a function of the unit price and a number of other determinants of demand. The demand function studies the relation between quantity demanded and unit price holding all the other determinants of demand constant. There is a concept of price-elasticity of demand that measures the relative logarithmic derivative of quantity demanded with respect to price.

The relevance of the discussion here is that the price-elasticity of demand depends on the values of the other determinants of demand that we are holding constant. In other words, a change in the value of one of the other determinants of demand could affect the price-elasticity of demand at a particular unit price. For instance, a change in an individual's income could affect the price-elasticity of demand function. Similarly, a change in the price of a substitute good could affect the price-elasticity of demand function.

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Partial truth and falsehood

Second-order mixed partial

The second-order mixed partial derivative captures precisely this fact. Basically, the second-order mixed partial derivative with respect to two of the input variables describes how the partial derivative with respect to one variable changes in terms of the second variable. The statement here can thus be interpreted as saying that the second-order mixed partial derivative of a function is not always zero.

Conditions where the value depends only on the specific input

The only cases where the partial derivative with respect to one variable depends only on that variable is where the function is additively separable in terms of a function purely of that variable and a function of the other variables. Another way of thinking of this is that the second-order mixed partials with respect to that particular variable and all the other variables are zero. Technically, the function needs to be a partially additively separable function between that variable and all the other variables.

For a function of two variables, this is the same as being an additively separable function between the two variables. Explicitly, for a function of the form:

$\!F(x,y):=f(x)+g(y)$

the partial derivative with respect to $x$ is just $f'(x)$, which does not depend on $y$, whereas the partial derivative with respect to $y$ is just $g'(y)$, which does not depend on $x$.

Here is an example of a function of three variables:

$f(x_{1},x_{2},x_{3}):=x_{2}^{2}-x_{1}^{2}x_{3}$

This function is a sum of a function purely of $x_{2}$ and a function that does not involve $x_{2}$. The partial derivative with respect to $x_{2}$ thus involves only $x_{2}$:

$f_{x_{2}}(x_{1},x_{2},x_{3})=2x_{2}$

On the other hand, because $f$ does not have this form with respect to $x_{1}$, the first partial derivative with respect to $x_{1}$ does depend on other variables. Specifically, it depends on both $x_{1}$ and $x_{3}$ in this case:

$f_{x_{1}}(x_{1},x_{2},x_{3})=-2x_{1}x_{3}$

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Related ideas

• Meaning of partial derivative depends on entire coordinate system