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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
dx
dy
dy
dz
dx
dz
)(xyy
Implicit differentiation
• From this we can take the fact that for a function of y:
dx
dyyf
dy
dxf
dx
d ))(())((
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 ydx
dx
dx
dy
dx
dx
dx
dy
dx
d
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:
dx
dyy
dx
dyy
dy
dy
dx
d 233 3)()(
dx
dyy
dx
dyy
dy
dy
dx
d 344 4)()(
dx
dy
dx
dyy
dy
dy
dx
d2)2()2(
Implicit differentiation: example 1
• Then we can apply this to the original function:
• Equals:
)2()()()()( 343 ydx
dx
dx
dy
dx
dx
dx
dy
dx
d
dx
dyx
dx
dyy
dx
dyy 23413 232
Implicit differentiation: example 1
• Next, we can rearrange to collect the terms:
• Equals:
dx
dy
dx
dyx
dx
dyy
dx
dyy 23413 232
dx
dyyyx )234(13 232
Implicit differentiation: example 1
• Finally:
• Equals:
234
1323
2
yy
x
dx
dy
dx
dyyyx )234(13 232
Implicit differentiation: example 2
• Example:
• Differentiate with respect to x
0sin4cos ydx
dxy
dx
d
0sin2cos 2 yxy
Implicit differentiation: example 2
• Then we can see, using the chain rule:
• And:
dx
dyy
dx
dyy
dy
dy
dx
dsin)(coscos
dx
dyy
dx
dyy
dy
dy
dx
dcos)(sinsin
Implicit differentiation: example 2
• We can then apply these to the function:
• Equals:
0sin4cos ydx
dxy
dx
d
0cos4sin dx
dyyx
dx
dyy
Implicit differentiation: example 2
• Now collect the terms:
• Which then equals:
dx
dyyyx )cos(sin4
dx
dy
yy
x
dx
dy
cossin
4
Implicit differentiation: example 3
• Example:
• Differentiate with respect to x
22322 yyxx
22322
dx
dy
dx
dyx
dx
dx
dx
d
Implicit differentiation: example 3
• To solve:
• We also need to use the product rule.
)()( 233232 x
dx
dyy
dx
dxyx
dx
d
32 yxdx
d
xydx
dyyx 2.3. 322
Implicit differentiation: example 3
• We can then apply these to the function:
• Equals:
22322
dx
dy
dx
dyx
dx
dx
dx
d
022.3.2 322 dx
dyyxy
dx
dyyxx
Implicit differentiation: example 3
• Now collect the terms:
• Which then equals:
dx
dy
dx
dyyxyxxy )32()22( 223
22
3
32
22
yxy
xxy
dx
dy
Implicit differentiation: example 4
• Example:
• Differentiate with respect to x
xyyx 2)arcsin( 22
xdx
dy
dx
dx
dx
d2)arcsin( 22
Implicit differentiation: example 4
• To solve:
• We also need to use the chain rule, and the fact that:
)arcsin( 2ydx
d
21
1)arcsin(
xx
dx
d
Implicit differentiation: example 4
• This is because:
, so
• Therefore:
• As x=sin(y), Therefore:
21
1)arcsin(
xx
dx
d
dx
dy
)arcsin(xy )sin(yx
22 1sin1cos xyydy
dx
Implicit differentiation: example 4
• Using this, we can apply it to our equation :
)arcsin( 2ydx
d
dx
dy
yy
dx
dy
yy
422 1
1.2
)(1
1.2
Implicit differentiation: example 4
• We can then apply these to the function:
• Equals:
xdx
dy
dx
dx
dx
d2)arcsin( 22
21
22
4
dx
dy
y
yx
Implicit differentiation: example 4
• Now collect the terms:
• Which then equals:
dx
dy
dx
dy
y
yx
41
222
y
yx
dx
dy
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:
dx
dyyf
dy
dxf
dx
d ))(())((