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.