2.7. DERIVATIVES OF FUNCTIONS GIVEN IMPLICITLY
141
the circle through a single function of the form y = f (x). At the same time, portions of
the circle can be represented explicitly as a function of x, such as the highlighted arc that
is magnified in the center of Figure 2.14. Moreover, it is evident that the circle is locally
linear, so we ought to be able to find a tangent line to the curve at every point; thus, it
makes sense to wonder if we can compute
dy
dx at any point on the circle, even though we
cannot write y explicitly as a function of x. Finally, we note that the righthand curve
in Figure 2.14 is called a lemniscate and is just one of many fascinating possibilities for
implicitly given curves.
In working with implicit functions, we will often be interested in finding an equation for
dy
dx that tells us the slope of the tangent line to the curve at a point (x, y). To do so, it will
be necessary for us to work with y while thinking of y as a function of x, but without being
able to write an explicit formula for y in terms of x. The following preview activity reminds
us of some ways we can compute derivatives of functions in settings where the function’s
formula is not known. For instance, recall the earlier example
d
dx [e u(x) ] = e u(x) u ′ (x).
Preview Activity 2.7. Let f be a differentiable function of x (whose formula is not known)
and recall that
d
dx [ f (x)] and f ′ (x) are interchangeable notations. Determine each of the
following derivatives of combinations of explicit functions of x, the unknown function f ,
and an arbitrary constant c.
(a)
d
dx
x
2 + f (x)
(b)
d
dx
x
2 f (x)
(c)
d
dx
c + x + f (x)
2
(d)
d
dx
f (x
2 )
(e)
d
dx
[x f (x) + f (cx) + c f (x)]
⊲⊳
Implicit Differentiation
Because a circle is perhaps the simplest of all curves that cannot be represented explicitly
as a single function of x, we begin our exploration of implicit differentiation with the
example of the circle given by x 2 + y 2 = 16. It is visually apparent that this curve is locally
linear, so it makes sense for us to want to find the slope of the tangent line to the curve
at any point, and moreover to think that the curve is differentiable. The big question is:
how do we find a formula for
dy
dx , the slope of the tangent line to the circle at a given
141
the circle through a single function of the form y = f (x). At the same time, portions of
the circle can be represented explicitly as a function of x, such as the highlighted arc that
is magnified in the center of Figure 2.14. Moreover, it is evident that the circle is locally
linear, so we ought to be able to find a tangent line to the curve at every point; thus, it
makes sense to wonder if we can compute
dy
dx at any point on the circle, even though we
cannot write y explicitly as a function of x. Finally, we note that the righthand curve
in Figure 2.14 is called a lemniscate and is just one of many fascinating possibilities for
implicitly given curves.
In working with implicit functions, we will often be interested in finding an equation for
dy
dx that tells us the slope of the tangent line to the curve at a point (x, y). To do so, it will
be necessary for us to work with y while thinking of y as a function of x, but without being
able to write an explicit formula for y in terms of x. The following preview activity reminds
us of some ways we can compute derivatives of functions in settings where the function’s
formula is not known. For instance, recall the earlier example
d
dx [e u(x) ] = e u(x) u ′ (x).
Preview Activity 2.7. Let f be a differentiable function of x (whose formula is not known)
and recall that
d
dx [ f (x)] and f ′ (x) are interchangeable notations. Determine each of the
following derivatives of combinations of explicit functions of x, the unknown function f ,
and an arbitrary constant c.
(a)
d
dx
x
2 + f (x)
(b)
d
dx
x
2 f (x)
(c)
d
dx
c + x + f (x)
2
(d)
d
dx
f (x
2 )
(e)
d
dx
[x f (x) + f (cx) + c f (x)]
⊲⊳
Implicit Differentiation
Because a circle is perhaps the simplest of all curves that cannot be represented explicitly
as a single function of x, we begin our exploration of implicit differentiation with the
example of the circle given by x 2 + y 2 = 16. It is visually apparent that this curve is locally
linear, so it makes sense for us to want to find the slope of the tangent line to the curve
at any point, and moreover to think that the curve is differentiable. The big question is:
how do we find a formula for
dy
dx , the slope of the tangent line to the circle at a given