3.3 Partial Derivatives
A partial derivative of a multivariable function is defined in much the same way as the derivative of a univariate function. For a function of two variables (say, $x$ and $y$) we can proceed in the same way as above, comparing the value of the function at $f(x,y)$ as we change the values of $x$ and $y$ by small amounts. The building blocks of our analysis are the partial derivatives of the function, which measure how the output of the function changes when one variable is increased while the other(s) are held constant.
In the case of a function of two variables, $x$ and $y$, we can define the partial derivative “with respect to $x$” as $\partial f/\partial x$, where \({\partial f \over \partial x} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x, y) - f(x, y) \over \Delta x}\) Visually, this has the same interpretation as above, except now the two points are points along the surface plot of the function $f(x,y)$, as as $\Delta x \rightarrow 0$ the line is tangent to the surface, not just a curve:
[ See interactive graph online at https://www.econgraphs.org/graphs/math/multivariable_calculus/partial_derivative_x ]
The partial derivative with respect to $y$ is defined similarly: holding $x$ constant, it measures the rate at which $f(x,y)$ changes when $y$ increases by a $\Delta y$. In the limit as $\Delta y \rightarrow 0$, it may be represented as a line tangent to the surface plot of the function, pointing in the $y$ direction.
[ See interactive graph online at https://www.econgraphs.org/graphs/math/multivariable_calculus/partial_derivative_y ]
One helpful way of thinking about partial derivatives is as the derivative of the function implied by holding the other variables of a multivariable function constant. (In economics, this often means imposing a “ceteris paribus” assumption that all other variables are held constant.)
To see what this means visually in this case, we can plot the (two-dimensional) function $f(x | y = \overline y)$. For illustrative purposes, we can see side-by-side what this looks like in three dimensions and two dimensions:
[ See interactive graph online at https://www.econgraphs.org/graphs/math/multivariable_calculus/holding_y_constant ]
Indeed, if you look carefully, you can see that the two-dimensional graph is exactly the same as the graph of the intersection of the surface with the plane at $y = \overline y$.
Calculating partial derivatives
When taking the partial derivative of a multivariate function, we’re varying one input while holding the others constant. Therefore, we just follow all the same rules as for univariate calculus, but just treat all variables except the one of interest as if they were constants.
For example, with the univariate function $f(x) = 12x^{1 \over 2}$, by the exponent rule we have \({df \over dx} = \tfrac{1}{2}\times 12x^{\frac{1}{2} - 1} = 6x^{-{1 \over 2}}\) For the multivariable function $f(x,y) = 4x^{1 \over 2}y$, the $y$ is treated as a constant when taking the partial derivative: \({\partial f \over \partial x} = \tfrac{1}{2}\times 4x^{\frac{1}{2} - 1}y = 2x^{-{1 \over 2}}y\) It’s easy to see that when $y = 3$, these two expressions are identical.