Find the derivative of y= 2x^-1^3/3x+1
Answers
Step-by-step explanation:
You can use the chain rule to find the derivative of this function by using
u
=
2
x
3
−
3
x
−
4
.
The chain rule tells you that you can differentiate a function
y
that depends on a variable
u
, which in turn depends on another variable
x
, by
d
d
x
(
y
)
=
d
d
u
(
y
)
⋅
d
d
x
(
u
)
Two additional rules of derivation, the power rule and the sum rule will come into play at certain points in the derivation of
y
.
So, if you use
u
=
2
x
3
−
3
x
−
4
, then you have
y
=
√
u
=
u
1
2
So, the derivative of
y
will look like this
y
′
=
d
d
u
(
y
)
⋅
d
d
x
(
u
)
y
′
=
d
d
u
(
u
1
2
)
⋅
d
d
x
(
2
x
3
−
3
x
−
4
)
The power rule tells you that you can differentiate a variable
x
raised to a power
a
by
x
a
=
a
⋅
x
a
−
1
This means that you have
d
d
u
(
u
1
2
)
=
1
2
⋅
u
1
2
−
1
d
d
u
(
u
1
2
)
=
1
2
⋅
1
u
1
2
=
1
2
⋅
1
√
u
The sum rule tells you that the derivative of a function that can be written as a sum of two (or more) functions is equal to the sum of the derivatives of those functions.
For
y
=
f
(
x
)
+
g
(
x
)
+
h
(
x
)
+
...
you have
d
d
x
(
y
)
=
f
′
(
x
)
+
g
′
(
x
)
+
h
′
(
x
)
+
...
This means that you can write
d
d
x
(
2
x
3
−
3
x
−
4
)
=
d
d
x
(
2
x
3
)
+
d
d
x
(
−
3
x
)
+
d
d
x
(
−
4
)
d
d
x
(
2
x
3
−
3
x
−
4
)
=
2
⋅
3
x
3
−
2
−
3
+
0
d
d
x
(
2
x
3
−
3
x
−
4
)
=
6
x
2
−
3
Your original derivative now becomes
y
′
=
1
2
⋅
1
√
u
⋅
(
6
x
2
−
3
)
y
]
=
1
2
⋅
1
√
(
2
x
3
−
3
x
−
4
)
⋅
3
(
2
x
2
−
1
)
Finally,
y
′
=
3
2
⋅
2
x
2
−
1
√
2
x
3
−
3
x
−
4