Suppose the equation
defines the relation between variables
and
.
Thinking of
as the independent variable, we apply the chain rule to differentiate the
equation and then solve the resulting equation for the derivative
.
This process is called **implicit differentiation**. The process
results in the equation

Exercise: Find the equation of the tangent line to the graph of the equation at the given points:

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

If
satisfies the equation
,
then
.
This is the result of the chain rule
.
To find the second derivative
,
apply the formula

i.e.,
replace
and
in
by
. To find the third derivative
,
replace
in
by
and
by