differentiation of y = x^2+ x^3+ 2x / x
Answers
Explanation:
Method 1 Use the product ruel then simplify.
The product rule tells us that the derivative of a product of two function (I think of them as the First and the Second) is given by:
d
d
x
(
F
S
)
=
F
'
S
+
F
S
'
(Because both addition and multiplication of functions are commutative, other orders are possible.)
So we get (including detail you might prefer to leave out eventually)
y
=
x
2
(
2
x
+
3
)
y
'
=
[
d
d
x
(
x
2
)
]
(
2
x
+
3
)
+
x
2
[
d
d
x
(
2
x
+
3
)
]
(usually we'll omit writing this step, but we need to DO this)
y
'
=
[
2
x
]
(
2
x
+
3
)
+
x
2
[
2
]
=
4
x
2
+
6
x
+
2
x
2
=
6
x
2
+
6
x
Method 2 Multiply first, the differentiate.
y
=
x
2
(
2
x
+
3
)
y
=
2
x
3
+
3
x
2
(by algebra)
Now we do not need the product rule, just the sum and power and constant multiple ruel)
y
'
=
6
x
2
+
6
x
Two lessons:
We can use either method to get to the correct answer. (There are many paths to one destination.)
We can take control of how a problem is written. (Unless our tester has told us we must use a particular method -- that is sometimes done to test our knowledge of that method.)