Up to this point, all of the equations that we have differentiated have had one variable (e.g., \(y\), \(f(x)\), or \(P\)) defined explicitly with respect to another variable (e.g., \(x\) or \(t\)).

For example, the equation \( y= 3x^2 - 2x + 1\) has \( y \) explicitly defined in terms of \( x \).

But how would we differentiate an equation, such as \(x^2+y^2=16\), where the variables are defined implicitly with respect to one another?

This equation represents a circle centred at \((0,0)\) with radius \(4\) units and relates the two variables \(x\) and \(y\).

Let's think about this problem geometrically.

It seems clear that a tangent line exists at each point on the circle. But does this mean a derivative \(\dfrac{dy}{dx}\) exists?

If so, how can we determine how it depends on \(x\)?

We could rearrange this equation to define one variable in terms of another.

Rearranging this equation results in two explicitly defined functions: \(y=\sqrt{16-x^2}\) and \(y=-\sqrt{16-x^2}\), representing the upper and lower semi-circles.

Using the differentiation techniques that we have learned so far, we could differentiate both functions separately.

For example, \(y=\sqrt{16-x^2}=(16-x^2)^{\frac{1}{2}}\) has the derivative

Thus, for the upper circle, \(\dfrac{dy}{dx}\) exists for \(-4 \lt x \lt 4\).

However, \(\dfrac{dy}{dx}\) does not exist at \(x=4\) and \(x=-4\), reflecting the fact that the tangent lines are vertical at those points.

But what do we do for an equation, such as \(x^4+12xy^3+4y^4=20\), which cannot be easily solved for \(y(x)\)?

Fortunately, there is a more general way to differentiate implicit equations.