Implicit Differentiation
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Transcript of Implicit Differentiation
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www.le.ac.uk
Implicit Differentiation
Department of MathematicsUniversity of Leicester
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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
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What is it?• An Implicit function is a function of the
form:
• Example:
0),( yxf
0343 xyxy
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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
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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.
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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
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Implicit differentiation• From this we can take the fact that for a
function of y:
dxdyyf
dydxf
dxd
))(())((
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Implicit differentiation: example 1• We can now use this to solve implicit
differentials
• Example:
yxyxy 2343
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Implicit differentiation: example 1• Then differentiate all the terms with
respect to x:
)2()()()()( 343 ydxdx
dxdy
dxdx
dxdy
dxd
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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.
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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(
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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
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Implicit differentiation: example 1• Next, we can rearrange to collect the
terms:
• Equals:
dxdy
dxdyx
dxdyy
dxdyy 23413 232
dxdyyyx )234(13 232
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Implicit differentiation: example 1• Finally:
• Equals:
2341323
2
yy
xdxdy
dxdyyyx )234(13 232
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Implicit differentiation: example 2• Example:
• Differentiate with respect to x
0sin4cos ydxdxy
dxd
0sin2cos 2 yxy
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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
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Implicit differentiation: example 2• We can then apply these to the function:
• Equals:
0sin4cos ydxdxy
dxd
0cos4sin dxdyyx
dxdyy
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Implicit differentiation: example 2• Now collect the terms:
• Which then equals:
dxdyyyx )cos(sin4
dxdy
yyx
dxdy
cossin4
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Implicit differentiation: example 3• Example:
• Differentiate with respect to x
22322 yyxx
22322
dxdy
dxdyx
dxdx
dxd
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Implicit differentiation: example 3• To solve:
• We also need to use the product rule.
)()( 233232 xdxdyy
dxdxyx
dxd
32 yxdxd
xydxdyyx 2.3. 322
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Implicit differentiation: example 3• We can then apply these to the function:
• Equals:
22322
dxdy
dxdyx
dxdx
dxd
022.3.2 322 dxdyyxy
dxdyyxx
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Implicit differentiation: example 3• Now collect the terms:
• Which then equals:
dxdy
dxdyyxyxxy )32()22( 223
22
3
3222yxyxxy
dxdy
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Implicit differentiation: example 4• Example:
• Differentiate with respect to x
xyyx 2)arcsin( 22
xdxdy
dxdx
dxd 2)arcsin( 22
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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
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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
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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
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Implicit differentiation: example 4• We can then apply these to the function:
• Equals:
xdxdy
dxdx
dxd 2)arcsin( 22
21
224
dxdy
y
yx
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Implicit differentiation: example 4• Now collect the terms:
• Which then equals:
dxdy
dxdy
y
yx41
222
yyx
dxdy
2)1)(22( 4
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Conclusion• Explicit differentials are of the form:
• Implicit differentials are of the form:
)(xfy
0),( yxf
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Conclusion• To differentiate a function of y with respect
to x we can use the chain rule:
dxdyyf
dydxf
dxd
))(())((
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