2. y x e tan x
4 3x
ln y ln x ln e ln tan x
4 3x
2
dy 1 4 sec x
3
dx y x tan x
2
dy 4 sec x
x e tan x{ 3
4 3x
}
dx x tan x
,3. y x sin 2 x cos 4 x
5
ln y ln x ln sin 2 x ln cos 4 x
5
dy 1 1 1 1
5 .5 x
4
2 cos 2 x . 4sin
dx y x sin 2 x cos 4 x
dy 1 1 1
y{ 5 .5 x
4
2 cos 2 x . 4si
dx x sin 2 x cos 4 x
dy 5
x sin 2 x cos 4 x{ 2 cot 2 x 4 tan 4 x}
5
dx x
, Implicit function
If y=𝒙𝟐 − 𝟒𝒙 + 𝟐, 𝒚 is completely defined in terms of x
y is called an explicit function of x.
When the relationship between x and y is more involve
may not be possible(or desirable) to separate y compl
on the LHS , e.g. xy+siny =2. In such a case as this
called an implicit function of x, because a relation
of the form y=f(x) is implied in the given equation.
it may still be necessary to determine the derivat
of y with respect to x and in fact this is not at all
, All we have to remember is that y is a functio
x, even if it is difficult to see what it is. I
fact, this is really an extension of our ‘funct
of a function’ routine.
𝑥 2 +𝑦 2 =25, as it stands is an example of an
implicit function.
Once again, all we have to remember is that y i
2 2 𝑑𝑦
function of x. so, if 𝑥 +𝑦 =25, let us find
21-Jun-20 MTH 121- General Mathematics II (Calculus)
𝑑
4 3x
ln y ln x ln e ln tan x
4 3x
2
dy 1 4 sec x
3
dx y x tan x
2
dy 4 sec x
x e tan x{ 3
4 3x
}
dx x tan x
,3. y x sin 2 x cos 4 x
5
ln y ln x ln sin 2 x ln cos 4 x
5
dy 1 1 1 1
5 .5 x
4
2 cos 2 x . 4sin
dx y x sin 2 x cos 4 x
dy 1 1 1
y{ 5 .5 x
4
2 cos 2 x . 4si
dx x sin 2 x cos 4 x
dy 5
x sin 2 x cos 4 x{ 2 cot 2 x 4 tan 4 x}
5
dx x
, Implicit function
If y=𝒙𝟐 − 𝟒𝒙 + 𝟐, 𝒚 is completely defined in terms of x
y is called an explicit function of x.
When the relationship between x and y is more involve
may not be possible(or desirable) to separate y compl
on the LHS , e.g. xy+siny =2. In such a case as this
called an implicit function of x, because a relation
of the form y=f(x) is implied in the given equation.
it may still be necessary to determine the derivat
of y with respect to x and in fact this is not at all
, All we have to remember is that y is a functio
x, even if it is difficult to see what it is. I
fact, this is really an extension of our ‘funct
of a function’ routine.
𝑥 2 +𝑦 2 =25, as it stands is an example of an
implicit function.
Once again, all we have to remember is that y i
2 2 𝑑𝑦
function of x. so, if 𝑥 +𝑦 =25, let us find
21-Jun-20 MTH 121- General Mathematics II (Calculus)
𝑑
