example: sec 3.7: implicit differentiation. example: in some cases it is possible to solve such an...

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