In many examples, especially the ones derived from differential
equations, the variables involved are not linked to each other in
an explicit way. Most of the time, they are linked through an
implicit formula, like F(x,y) =0. Once x is fixed, we may
find y through numerical computations. (By some fancy theorems,
we may formally show that y may indeed be seen as a function of
x over a certain interval). The question becomes what is the
at least at a certain a
point? The method of implicit differentiation answers
this concern. Let us illustrate this through the following
Example. Find the equation of the tangent line to the
25 x2 + y2 = 109
at the point (2,3). One way is to find y as a function of
x from the above equation, then differentiate to find the slope
of the tangent line. We will leave it to the reader to do the
details of the calculations. Here, we will use a different
method. In the above equation, consider y as a
function of x:
25 x2 + y(x)2 = 109,
and use the techniques of differentiation, to get
From this, we obtain
which implies that
at the point (2,3). So the equation of the tangent line is
3y + 50 x = 109.
You may wonder why bother if this is just a different way of
finding the derivative? Consider the following example! It can be
very hard or in fact impossible to solve explicitly for y as a
function of x.
Example. Find y' if
This is a wonderful example of an implicit relation between xand y. So how do we find y'? Let us differentiate the above
equation with respect to x where y is considered to be a
function of x. We get
Easy algebraic manipulations give
We can also find higher derivatives of y such as y'' in this
manner. We only have to differentiate the above result. Of
course the calculations get little more messy.
Exercise 1. Find y' if
xy3 + x2y2 + 3x2 - 6 = 1.
Exercise 2. Prove that an equation of the tangent line to
the graph of the hyperbola
at the point
Exercise 3. Show that if a normal line to each point on an
ellipse passes through the center of an ellipse, then the ellipse
is a circle.
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