Introduction
What do we mean by "implicit differentiation"?
When we have $y$ explicitly
defined as a function of $x$, say $y=f(x)=x^2$, we can find
$\frac{dy}{dx}$ by differentiating $x^2$. In other
circumstances, we know $y=f(x)$ is a function of $x$, but we do
not know what $f$ is, so we say that $y$ is implicitly defined as a function of
$x$. In this situation, to differentiate $y$, we get
$\frac{d}{dx}(y)=\frac{dy}{dx}$ (of course, these expressions are
the same), because we do not know what function to
differentiate.
Why we care
In addition to taking derivatives of functions
we can take derivatives of equations.
After all, if both sides of an equation are always equal, then
their rates of change are also equal. If the equation
involves both $x$ and $y$, and we are differentiating with respect
to $x$, then $\frac{d}{dx}(y)=\frac{dy}{dx}$ as explained above,
because that is the best we can do with the derivative of $y$
without knowing the formula for $y$.
Notice that we have curves for which $y$ is not explicitly a
function of $x$, such as the unit circle $x^2+y^2=1$. Here,
if we solve for $y$, we get $y=\pm\sqrt{1x^2}$ which is not a
function. (Why not? Think about this
before reading more. To see why: Plug in a
value for $x$, say $x=0$, and you get two values for $y$, namely
$1$ and $1$. You can also see this curve does not represent
a function by looking at the vertical line test.) However,
we can still find $\frac{dy}{dx}$, by using implicit
differentiation, by using the values of $x$ and $y$, not
just $x$, to decide where we are differentiating. This
allows us to use the valid function, either $y=\sqrt{1x^2}$ or
$y=\sqrt{1x^2}$ depending on whether $y$ is positive or
negative.
