# Implicit differentiation

So far we’ve only differentiated functions, or relations where $\displaystyle y$ was written *explicitly* as a function of $\displaystyle x$. We can
also differentiate relations that aren’t functions, and they are often written *implicitly* (sometimes there is no other way). To differentiate these relations, we take the derivative of both sides and solve for the derivative that we want.

## Example

Consider the following relation:

$\displaystyle x^{2} + y^{2} = 25$.

This describes a circle centred about the origin with a radius of 5. In this relation, $\displaystyle y$ is not a function of $\displaystyle x$, but we still want to find the derivative of the former with respect to the latter. We start by taking the derivative of both sides:

$\displaystyle \frac{d}{dx} \left ( x^{2} + y^{2} \right ) = \frac{d}{dx} 25$.

We can apply the sum rule on the left and the constant rule on the right:

$\displaystyle \frac{d}{dx} x^{2} + \frac{d}{dx} y^{2} = 0$.

The $\displaystyle x^{2}$ term is easy, but we need the chain rule for the $\displaystyle y^{2}$ term:

$\displaystyle 2 x + 2 y \cdot \frac{dy}{dx} = 0$.

Now we can simply solve for the derivative that has just appeared:

$\displaystyle \frac{dy}{dx} = - \frac{x}{y}$.

We’re done! That’s all there is to it. Try verifying this derivative by looking at the tangent slopes on a circle with a radius of 25.