The document contains examples of differentiation formulas and their applications: 1) It provides the formula for differentiating y=u/v with respect to x and works through an example. 2) It differentiates y=x^2 - 5/2x^2 with respect to x using the formula. 3) Additional examples include differentiating trigonometric, exponential, and other functions with respect to x.

2.  Examples
 So one thing divided by another.
2
3
2 4x
x
+

3.  y = u/v
 To differentiate use the formula

2
du dv
v -u
dy dx dx=
dx v

4. 2
du
x
dx
=
u = x2
- 5
v= 2x2
4
dv
x
dx
=
2 2
2 2
(2 )(2 ) ( 5)(4 )
(2 )
dy x x x x
dx x
− −
=
3 3
4
4 4 20
4
dy x x x
dx x
− +
= 3
5dy
dx x
=
2
2
x -5
y=
2x

5. 1
Differentiate with respect to x.
2+5x
u=1
du
=0
dx
v=2+5x
dv
=5
dx
2 2
dy (2+5x)0-1(5) -5
= =
dx (2+5x) (2+5x)

6. 3
3
4
2
1.
2 4
2 3
2.
4
cos
3.
4.
3 1
3 3
5.
sin
6.
2
x
y
x
x x
y
x
x
y
x
x
y
x
x
t
x
x
y
x
=
+
+
=
+
=
=
+
+
=
=

7. 3
3
4
2
1.
2 4
2 3
2.
4
cos
3.
4.
3 1
3 3
5.
sin
6.
2
x
y
x
x x
y
x
x
y
x
x
y
x
x
t
x
x
y
x
=
+
+
=
+
=
=
+
+
=
=