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Title: DIFFERENTIATION EXAM LECTURE NOTES 101
Description: differentiation lecture notes for engineering, yale university
Description: differentiation lecture notes for engineering, yale university
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DIFFERENTION
Is the Process of finding derivative
Is used to determine the rate of change with respect to time
...
dy/dx = y2-y1/x2-x1 =gradient of the graph note that
dy/dx is the derivative of y with respect to x
DIFFERENTIATION FROM THE FIRST PRINCIPLE
Consider the function f(x) that depend on a variable x
F(x)=3x2
then f(x)=6x
DIFFERENTIAL TABLE
y
dy/dx
constant
0
Sin x
Cos x
Cos x
-sin x
ex
ex
ekx
K ekx
Tan x
Sec2x
In |x|
1/x
Sin ax
a cos ax
Cos ax
-a sinax
METHODS OF DIFFERENTIATION
1
...
Product rule
3
...
Y=f[g(x)]
Let u be g(x)
Example
Y=f(x)= (3 + x2)3
Solution: let u be ((3 + x2)3
Y=u3
dy/du=3u2
du/dx=2x dy/dx
= dy/du × du/dx
3u2 × 2x but u = 3 + x2
= 6x (3 + x2 )2
PRODUCT RULE
We consider finding derivative of f(x) that can be written as a product of two
functions
...
Let y be f(x) = u(x)/v(x)
=u/v
(u/v)’= u(1/v) ‘ + 1/v(u)’
= (vu’ – uv’)/v2
Example
Y= sin x/x Solution:
U=sin x
v= x
U’= cos x
v’= 1
(X (cos x ) – sin x)/x2
Example 2
Y= cos x/x2 Solution:
U= cos x
U’= -sin x
v = x2
v’= 2x
(Vu’ – uv’)/v2
(X2(-sin x) – cos x (2x))/x4
(-x2sin x – 2x(cos x))/x4
= [ sin x/x2 – 2 cos x/x3]
DIFFERENTIAL TABLE
y
dy/dx
constant
0
Sin x
Cos x
Cos x
-sin x
ex
ex
ekx
K ekx
Tan x
Sec2x
In |x|
1/x
Sin ax
a cos ax
Cos ax
-a sinax
METHODS OF DIFFERENTIATION
1
...
Product rule
3
...
Y=f[g(x)]
Let u be g(x)
Example
Y=f(x)= (3 + x2)3
Solution: let u be ((3 + x2)3
Y=u3
dy/du=3u2
du/dx=2x dy/dx
= dy/du × du/dx
3u2 × 2x but u = 3 + x2
= 6x (3 + x2 )2
PRODUCT RULE
We consider finding derivative of f(x) that can be written as a product of two
functions
...
Let y be f(x) = u(x)/v(x)
=u/v
(u/v)’= u(1/v) ‘ + 1/v(u)’
= (vu’ – uv’)/v2
Example
Y= sin x/x Solution:
U=sin x
v= x
U’= cos x
v’= 1
(X (cos x ) – sin x)/x2
Example 2
Y= cos x/x2 Solution:
U= cos x
U’= -sin x
v = x2
v’= 2x
(Vu’ – uv’)/v2
(X2(-sin x) – cos x (2x))/x4
(-x2sin x – 2x(cos x))/x4
= [ sin x/x2 – 2 cos x/x3]
Title: DIFFERENTIATION EXAM LECTURE NOTES 101
Description: differentiation lecture notes for engineering, yale university
Description: differentiation lecture notes for engineering, yale university