How to do implicit differentiation (in Python, using SymPy)

Task

Assume we have an equation in which $y$ cannot be isolated as a function of $x$. (The standard example is the formula for the unit circle, $x^2+y^2=1$.) We would still like to be able to compute the derivative of $y$ with respect to $x$.

Related tasks:

How to graph curves that are not functions

Solution

This answer assumes you have imported SymPy as follows.

from sympy import *                   # load all math functions
init_printing( use_latex='mathjax' )  # use pretty math output

Let’s consider the example of the unit circle, $x^2+y^2=1$.

To plot it, SymPy first expects us to move everything to the left-hand side of the equation, so in this case, we would have $x^2+y^2-1=0$.

We then use that left hand side to represent the equation as a single formula, and computue $\frac{dy}{dx}$ using the idiff function (standing for “implicit differentiation”).

var( 'x y' )
formula = x**2 + y**2 - 1   # to represent x^2+y^2=1
idiff( formula, y, x )

$\displaystyle - \frac{x}{y}$

So in this case, $\frac{dy}{dx}=-\frac xy$.

Content last modified on 24 July 2023.

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Contributed by Nathan Carter (ncarter@bentley.edu)