find the partial derivatives with respect.to x of the xy² -5y+6.
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Step-by-step explanation:
Assuming that we want to find the derivative with respect to
x
of
x
y
2
(assumong that
y
is a function of
x
:
First use the product rule:
d
d
x
(
x
y
2
)
=
d
d
x
(
x
)
y
2
+
x
d
d
x
(
y
2
)
Now for
d
d
x
(
y
2
)
we'll need the power and chain rules.
d
d
x
(
x
y
2
)
=
1
y
2
+
x
[
2
y
d
y
d
x
]
d
d
x
(
x
y
2
)
=
y
2
+
2
x
y
d
y
d
x
If you want to differentiate the expression with respect to
t
then the derivatives above are all
d
d
t
and
d
x
d
t
may not be
1
.
d
d
t
(
x
y
2
)
=
y
2
d
x
d
t
+
2
x
y
d
y
d
t
If you want the partial derivatives of the function
f
(
x
,
y
)
=
x
y
2
that has been answered in another answer.
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