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Implicit Differentiation

Department of MathematicsUniversity of Leicester

What is it?• The normal function we work with is a

function of the form:

• This is called an Explicit function.Example:

)(xfy

32 xy

What is it?• An Implicit function is a function of the

form:

• Example:

0),( yxf

0343 xyxy

What is it?• This also includes any function that can be

rearranged to this form: Example:

• Can be rearranged to:

• Which is an implicit function.

yxyxy 2343

02343 yxyxy

What is it?• These functions are used for functions that

would be very complicated to rearrange to the form y=f(x).

• And some that would be possible to rearrange are far to complicated to differentiate.

Implicit differentiation• To differentiate a function of y with respect

to x, we use the chain rule.• If we have: , and

• Then using the chain rule we can see that:

)(yzz

dxdy

dydz

dxdz

)(xyy

Implicit differentiation• From this we can take the fact that for a

function of y:

dxdyyf

dydxf

dxd

))(())((

Implicit differentiation: example 1• We can now use this to solve implicit

differentials

• Example:

yxyxy 2343

Implicit differentiation: example 1• Then differentiate all the terms with

respect to x:

)2()()()()( 343 ydxdx

dxdy

dxdx

dxdy

dxd

Implicit differentiation: example 1• Terms that are functions of x are easy to

differentiate, but functions of y, you need to use the chain rule.

Implicit differentiation: example 1• Therefore:

• And:

dxdyy

dxdyy

dydy

dxd 233 3)()(

dxdyy

dxdyy

dydy

dxd 344 4)()(

dxdy

dxdyy

dydy

dxd 2)2()2(

Implicit differentiation: example 1• Then we can apply this to the original

function:

• Equals:

)2()()()()( 343 ydxdx

dxdy

dxdx

dxdy

dxd

dxdyx

dxdyy

dxdyy 23413 232

Implicit differentiation: example 1• Next, we can rearrange to collect the

terms:

• Equals:

dxdy

dxdyx

dxdyy

dxdyy 23413 232

dxdyyyx )234(13 232

Implicit differentiation: example 1• Finally:

• Equals:

2341323

2

yy

xdxdy

dxdyyyx )234(13 232

Implicit differentiation: example 2• Example:

• Differentiate with respect to x

0sin4cos ydxdxy

dxd

0sin2cos 2 yxy

Implicit differentiation: example 2• Then we can see, using the chain rule:

• And:

dxdyy

dxdyy

dydy

dxd sin)(coscos

dxdyy

dxdyy

dydy

dxd cos)(sinsin

Implicit differentiation: example 2• We can then apply these to the function:

• Equals:

0sin4cos ydxdxy

dxd

0cos4sin dxdyyx

dxdyy

Implicit differentiation: example 2• Now collect the terms:

• Which then equals:

dxdyyyx )cos(sin4

dxdy

yyx

dxdy

cossin4

Implicit differentiation: example 3• Example:

• Differentiate with respect to x

22322 yyxx

22322

dxdy

dxdyx

dxdx

dxd

Implicit differentiation: example 3• To solve:

• We also need to use the product rule.

)()( 233232 xdxdyy

dxdxyx

dxd

32 yxdxd

xydxdyyx 2.3. 322

Implicit differentiation: example 3• We can then apply these to the function:

• Equals:

22322

dxdy

dxdyx

dxdx

dxd

022.3.2 322 dxdyyxy

dxdyyxx

Implicit differentiation: example 3• Now collect the terms:

• Which then equals:

dxdy

dxdyyxyxxy )32()22( 223

22

3

3222yxyxxy

dxdy

Implicit differentiation: example 4• Example:

• Differentiate with respect to x

xyyx 2)arcsin( 22

xdxdy

dxdx

dxd 2)arcsin( 22

Implicit differentiation: example 4• To solve:

• We also need to use the chain rule, and the fact that:

)arcsin( 2ydxd

21

1)arcsin(x

xdxd

Implicit differentiation: example 4• This is because: , so• Therefore:

• As x=sin(y), Therefore:

21

1)arcsin(x

xdxd

dxdy

)arcsin(xy )sin(yx

22 1sin1cos xyydydx

Implicit differentiation: example 4• Using this, we can apply it to our

equation :

)arcsin( 2ydxd

dxdy

yy

dxdy

yy

422 1

1.2)(1

1.2

Implicit differentiation: example 4• We can then apply these to the function:

• Equals:

xdxdy

dxdx

dxd 2)arcsin( 22

21

224

dxdy

y

yx

Implicit differentiation: example 4• Now collect the terms:

• Which then equals:

dxdy

dxdy

y

yx41

222

yyx

dxdy

2)1)(22( 4

Conclusion• Explicit differentials are of the form:

• Implicit differentials are of the form:

)(xfy

0),( yxf

Conclusion• To differentiate a function of y with respect

to x we can use the chain rule:

dxdyyf

dydxf

dxd

))(())((