# Implicit differentiation

So far we’ve only differentiated functions, or relations where $y$ was written *explicitly* as a function of $x\text{.}$ 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:

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

This describes a circle centred about the origin with a radius of 5. In this relation, $y$ is not a function of $x\text{,}$ 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:

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

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

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

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

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

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

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

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.