Post on 02-Jan-2016
Example:123 xy
' : yFind
Example: 223 xyxy
' : yFind
Sec 3.7: Implicit Differentiation
Example:
123 xy
Example:
223 xyxy
In some cases it is possible to solve such an equation for as an explicit function
In many cases it is difficult (impossible to write y in terms of x)
Fortunately, we don’t need to solve an equation for y in terms of x in order to find the derivative of y . Instead we can use the method of implicit differentiation
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
Sec 3.7: Implicit Differentiation
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
'yydx
d
2ydx
d
Sec 3.7: Implicit Differentiation
Remember:y is a function of x and using the Chain Rule,
5ydx
d
yedx
d
ydx
dsin
'2yy
'5 4 yy
'ye y
'cos yy
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
'yydx
d
'22 yyydx
d
Example:
223 xyxy
' : yFind
Sec 3.7: Implicit Differentiation
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
')(cossin yyydx
d
Example:
xyyx cos)sin( 2 ' : yFind
Sec 3.7: Implicit Differentiation
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
'yydx
d
'22 yyydx
d
32
21
y
y
ydx
d
')(cossin yyydx
d
'yeedx
d yy
y
yy
dx
d
2
'
Sec 3.7: Implicit Differentiation
Remember:
In finding y’’ you may use the original equation
Example:
1644 yx '' : yFind
fat circle
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation
Sec 3.7: Implicit Differentiation