Add (2x3+7y3−8z3), (4x3−3y3+11z3) and (−6z3+2y3−9x3).
Answers
Answer:
⠀⠀⠀⣠⠴⠶⠦⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣤⡑⢄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣟⡛⠛⠻⢿⣿⣮⣧⠀⠀⠀
⠀⠀⣤⣤⣀⣀⡙⠿⠇⠀⠀⠀⠹⣿⣿⣇⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⠸⣿⣿⡀⠀
⠀⠀⣿⣿⠉⠉⠛⠻⣿⣿⣿⣿⣿⠀⢿⣿⡇⠀
⠀⠀⣿⣿⡄⠀⠀⠀⣿⣿⠀⠀⠈⠁⢸⣿⡇⠀
⠀⠀⢻⣿⣧⡀⠀⢀⣿⣿⣧⡀⠀⢀⣾⣿⠇⠀
⠀⠀⠈⢿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⡟⠀⠀
⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠙⠛⠛⠉⠀⠀⠀
⠀⠀⣼⣿⣿⣿⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢿⡿⠛⠿⠿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⣷⣄⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⣏⠉⠛⠛⠻⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣽⣶⣤⣄⣀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀
⠀⠀⠙⢧⡀⠈⠉⠉⠛⢻⡿⠿⣧⠀⠀⠀⠀⠀
⠀⠀⣴⣾⣿⣶⣶⣤⣤⣸⣇⡀⠀⠀⠀⠀⠀⠀
⠀⠀⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀
⠀⠀⠈⢳⣄⣀⣀⠀⠉⢹⡟⠛⠓⠀⠀⠀⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣶⣄⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⠋⢩⡉⠙⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⣿⠀⢸⣷⣀⡀⣸⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢻⣆⠈⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣶⣾⣷⣦⣍⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⣩⣿⣟⠛⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠿⠍⠥⠃⠀⠀
Step-by-step explanation:
Explanation:
Starting with:
4
x
3
+
3
y
3
=
6
Differentiate both sides with respect to
x
:
12
x
2
+
9
y
2
d
y
d
x
=
0
Solve for
d
y
d
x
(See Note 1 , below)
d
y
d
x
=
−
12
x
2
9
y
2
, so
d
y
d
x
=
−
4
x
2
3
y
2
Differentiate again, using the quotient rule to get
d
2
y
d
x
2
=
(
−
8
x
)
(
3
y
2
)
−
(
−
4
x
2
)
(
6
y
d
y
d
x
)
(
3
y
2
)
2
=
−
24
x
y
2
+
24
x
2
y
d
y
d
x
9
y
4
I prefer to remove the common factor before proceeding:
=
−
24
x
y
⎛
⎜
⎝
y
−
x
d
y
d
x
9
y
4
⎞
⎟
⎠
Now, replace
d
y
d
x
=
−
24
x
y
⎛
⎜
⎝
y
−
x
−
4
x
2
3
y
2
9
y
4
⎞
⎟
⎠
=
−
24
x
y
⎛
⎜
⎝
y
+
4
x
3
3
y
2
9
y
4
⎞
⎟
⎠
Now, simplify the complex fraction using your chosen technique.
=
−
24
x
y
⎛
⎜
⎜
⎝
(
y
+
4
x
3
3
y
2
)
(
3
y
2
)
(
9
y
4
)
(
3
y
2
)
⎞
⎟
⎟
⎠
=
−
24
x
y
(
3
y
3
+
4
x
3
27
y
6
)
I see that I can reduce the fraction, but before I do there's a step I can do to simplify a lot.
Way back at the start of the problem, we were told that
4
x
3
+
3
y
3
=
6
So the numerator of our fraction is
6
.
(See Note 2 below.)
=
−
24
x
y
(
6
27
y
6
)
Now simplify the quotient:
d
2
y
d
x
2
=
−
16
x
3
y
5
Note 1
Although we could differentiate again immediately, I prefer not to.
If we differentiate without solving for
d
y
d
x
first, we will need to be careful to disti nguish
(
d
y
d
x
)
2
from
d
2
y
d
x
2
. We get
24
x
+
18
y
d
y
d
x
d
y
d
x
+
9
y
2
d
2
y
d
x
2
=
0
.
It works, but it's kind of a mess.
Note 2
This step is typical of certain kinds of implicit differentiation second derivative problems. If you remember to look for it, it can simplify the result considerably.