The process of implicit differentiation is best understood by
looking at examples. You will see that when we treat $y$ as
a function of $x$, often we must use the chain
rule**.
Example: Find the equation of the tangent line to $x^2+y^2=1$
at $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.
Solution: First we find $\frac{dy}{dx}$ so we can find the
slope. We differentiate both sides of $x^2+y^2=1$ with respect
to $x$: $$ \frac{d}{dx}\left(x^2+y^2\right) = \frac{d}{dx}(1). $$
The derivative of $x^2$ is $2x$. The derivative of the constant 1 is
0. By the chain rule, $$ \frac{d}{dx}\left(y^2\right) = 2 y \cdot
\frac{dy}{dx}^{**}.$$ (**for example, say
$y=\frac{1}{x}$. Then
$\frac{d}{dx}(y^2)=\frac{d}{dx}\left(\frac{1}{x}\right)^2=2\left(\frac{1}{x}\right)^1\frac{d}{dx}\left(\frac{1}{x}\right)=2
y \cdot \frac{dy}{dx}$.) We now have
$$\begin{eqnarray*}2x + 2y \frac{dy}{dx} &=& 0\\
2y\frac{dy}{dx}&=&-2x\\ \frac{dy}{dx} &=&
-\frac{x}{y}. \end{eqnarray*}$$ We see that this answer involves
both $x$ and $y$, which is necessary to determine whether this
derivative refers to the top of the circle or the bottom.
To figure out the slope at $\left(\frac{\sqrt{3}}{2},
\frac{1}{2}\right)$ we just plug in these values to $\frac{dy}{dx}$
to get the slope $m= \frac{dy}{dx}\Big|_{\left(\frac{\sqrt{3}}{2},
\frac{1}{2}\right)} = -\frac{\sqrt3/2}{1/2}=- \sqrt{3}.$ Therefore,
the equation to the tangent line is $y-\frac{1}{2}=-\sqrt
3\left(x-\frac{\sqrt 3}{2}\right)$. DO:
Write this equation in slope intercept form. Graph this tangent
line along with the unit circle. Do the slope and
$y$-intercept make sense? What would happen at the point $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$?.
In the following video we set up the machinery of implicit
differentiation and work more examples.