Derivatives of Implicit Functions

Concept First: Explicit vs. Implicit Functions

So far you have differentiated functions in the form $y = f(x)$ — an explicit function, where $y$ is given directly in terms of $x$ (e.g. $y = x^2 + 3x$ or $y = \sin x$). But many relations are not in this solved form. Consider:

1. $x - y - \pi = 0$
2. $x + \sin(xy) - y = 0$

The first solves easily to $y = x - \pi$. In the second, solving for $y$ is not straightforward, yet $y$ still depends on $x$ — the relation just hides it. When $y$ is not isolated we call $y$ an implicit function of $x$. The goal here is to find $\frac{dy}{dx}$ directly from such a relation, without solving for $y$.

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Differentiating Implicit Functions: The Core Idea

Differentiate every term of the relation with respect to $x$, treating $y$ as a function of $x$ — so apply the chain rule to any term involving $y$. For example:

After differentiating every term, solve the resulting equation algebraically for $\frac{dy}{dx}$.