]>
Examples of using the differentiation rules:
ⅆ
ⅆ
x
(
3
x
2
+
x
)
=
ⅆ
ⅆ
x
(
3
x
2
)
+
ⅆ
ⅆ
x
(
x
)
by the sum rule
=
3
ⅆ
ⅆ
x
(
x
2
)
+
ⅆ
ⅆ
x
(
x
1
2
)
by the constant multiple rule and rewriting
=
3
⋅
2
x
+
1
2
x
−
1
2
by the power rule
=
6
x
+
1
2
x
Although we could expand the next function, it is easier to use the product
rule:
ⅆ
ⅆ
x
(
(
x
3
−
x
+
1
)
(
x
3
+
x
−
7
)
)
=
(
ⅆ
ⅆ
x
(
x
3
−
x
+
1
)
)
(
x
3
+
x
−
7
)
+
(
ⅆ
ⅆ
x
(
x
1
3
+
x
−
7
)
)
(
x
3
−
x
+
1
)
=
(
3
x
2
−
1
+
0
)
(
x
3
+
x
−
7
)
+
(
1
3
x
−
2
3
+
1
−
0
)
(
x
3
−
x
+
1
)
=
(
3
x
2
−
1
)
(
x
3
+
x
−
7
)
+
(
1
3
x
−
2
3
+
1
)
(
x
3
−
x
+
1
)
There is not much to be gained from expanding the expression above, so this is
an acceptable answer.
ⅆ
ⅆ
x
(
x
π
x
−
7
+
x
7
)
=
(
ⅆ
ⅆ
x
(
x
π
)
)
(
x
−
7
+
x
7
)
−
(
ⅆ
ⅆ
x
(
x
−
7
+
x
7
)
)
x
π
(
x
−
7
+
x
7
)
2
=
π
x
π
−
1
(
x
−
7
+
x
7
)
−
(
−
7
x
−
8
+
7
x
6
)
x
π
(
x
−
7
+
x
7
)
2
Again, since we are only trying to find the derivative, there is no need to do
any further simplification.