Worksheet ----MS125 Calculus I Dr. Y. Kim
3.8 Implicit Differentiation
What is an implicit function?
Function that can be written in the form )(xfy
=
is called an explicit function of x. However an
equation in x and y, such as 4
22
=+ yx , is said to give an implicit function of x. The graph of
4
22
=+ yx is the circle with the center )0,0( and the radius 2. Since there are x-values which
correspond to two y-values, y is not a function of x on the whole circle. But the equation represents
a curve which has a tangent line at each point. The slope of each tangent line can be found by
differentiating the equation with respect to x. This differentiation is called the implicit
differentiation
.
Tip:
Consider each y as a function of x.
Ex1) Compute the following
1.
[
]
3
4x
dx
d
2.
[
]
3
y
dx
d
3.
[ ]
yx
dx
d
32 +
4.
[
]
2
xy
dx
d
Ex2) Differentiate each implicit function (or Find
dx
dy
)
1. 13
2
=−
xy
2. 4
22
=+
yx
3. 5
=
xy
4. 125
323
=++
yxyx
5. 0
252
=+−
yxex y
6.
)cos(xyy
=
Ex3) Find the slope of the tangent line to the graph of the equation 4
22 =+ yx at the point
)3,1(
.
HOMEWORK
: 1~33 odd
