implicit differentiation objective: to find derivatives of functions that we cannot solve for y
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Implicit Differentiation
Objective: To find derivatives of functions that we cannot solve for y.
Explicit vs. Implicit
• An equation of the form y = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation.
Explicit vs. Implicit
• An equation of the form y = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation.
• Sometimes functions are defined by equations in which y is not alone on one side. For example
is not of the form y = f(x), but it still defines y as a function of x since it can be rewritten as
xyxy 1
1
1
x
xy
Explicit vs. Implicit
• We say that the first form of the equation
defines y implicitly as a function of x.
xyxy 1
Explicit vs. Implicit
• An equation in x and y can implicitly define more than one function in x. This can occur when the graph of the equation fails the vertical line test, so it is not the graph of a function.
Explicit vs. Implicit
• An equation in x and y can implicitly define more than one function in x. This can occur when the graph of the equation fails the vertical line test, so it is not the graph of a function.
• For example, if we solve the equation of the circle for y in terms of x, we obtain . This gives us two functions that are defined implicitly.
122 yx 21 xy
Explicit vs. Implicit
• Definition 3.1.1 • We will say that a given equation in x and y defines
the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation.
Implicit Differentiation
• It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.
Implicit Differentiation
• It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.
• For example, we can take the derivative of with the quotient rule:
1
1
x
xy
22 )1(
2
)1(
)1)(1()1)(1(
xx
xx
dx
dy
Implicit Differentiation
• We can also take the derivative of the given function without solving for x by using a technique called implicit differentiation.
Implicit Differentiation
• We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.
xyxy 1
Implicit Differentiation
• We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.
xyxy 1
1dx
dyy
dx
dyx y
dx
dyx 1)1(
Implicit Differentiation
• We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.
xyxy 1
1dx
dyy
dx
dyx y
dx
dyx 1)1(
1
1
x
y
dx
dy
1
1
x
xy
Implicit Differentiation
• We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.
xyxy 1
1dx
dyy
dx
dyx y
dx
dyx 1)1(
1
1
x
y
dx
dy
2)1(
2
111
11
xx
xx
xx
dx
dy
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
xdx
dyy
dx
dyy 2cos10
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
xdx
dyy
dx
dyy 2cos10
xdx
dyyy 2)cos10(
yy
x
dx
dy
cos10
2
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
y
x
dx
dy 2
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
22
2
222
yyx
xy
dx
yd
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
22
2
222
yyx
xy
dx
yd
3
22
2
2 42
y
xy
dx
yd
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
22
2
222
yyx
xy
dx
yd
3
22
2
2 42
y
xy
dx
yd
32
2 9
ydx
yd
Example 4
• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).012 xy
Example 4
• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).• We know that the slope of the tangent line means
the value of the derivative at the given points.
012 xy
Example 4• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).• We know that the slope of the tangent line means
the value of the derivative at the given points.
012 xy
012 dx
dyy
ydx
dy
2
1
Example 4• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).• We know that the slope of the tangent line means
the value of the derivative at the given points.
012 xy
012 dx
dyy
ydx
dy
2
1
2
1
12
yxdx
dy
2
1
12
yxdx
dy
Example 5
a) Use implicit differentiation to find dy/dx for the equation .
b) Find the equation of the tangent line at the pointxyyx 333
2
3,2
3
Example 5
a) Use implicit differentiation to find dy/dx for the equation .xyyx 333
][333 22 ydx
dyx
dx
dyyx
22 33)33( xydx
dyxy
xy
xy
xy
xy
dx
dy
2
2
2
2
33
33
Example 5
a) Use implicit differentiation to find dy/dx for the equation .
b) Find the equation of the tangent line at the pointxyyx 333
2
3,2
3
xy
xy
dx
dy
2
2
)2/3()2/3(
)2/3()2/3(2
2
dx
dy
14/3
4/3
46
49
49
46
dx
dy
3
)(1 23
23
xy
xy
Homework
• Section 3.1• Page 190• 1-19 odd• 19 (just use implicit)