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# Step-by-Step Math Answer

*
w
*
is a function of two variables,
*
x
*
and
*
y
*
.

Since
*
w
*
is a function of two variables,
*
x
*
and
*
y
*
, we find ∂
*
w
*
/ ∂
*
x
*
and ∂
*
w
*
/ ∂
*
y
*
:

*
x
*
and
*
y
*
are functions of two variables,
*
r
*
and
*
θ
*
.

*
x
*
and
*
y
*
are functions of two variables,
*
r
*
and
*
θ
*
, so we find ∂
*
x
*
/ ∂
*
r
*
, ∂
*
x
*
/ ∂
*
θ
*
, ∂
*
y
*
/ ∂
*
r
*
, and ∂
*
y
*
/ ∂
*
θ
*
:

Find ∂
*
w
*
/ ∂
*
r
*
by applying the Chain Rule for two independent variables.

We find ∂
*
w
*
/ ∂
*
r
*
by applying the Chain Rule for two independent variables:

Substitute the expressions for
*
x
*
and
*
y
*
.

Substituting the expressions for
*
x
*
and
*
y
*
, we have

Factoring out
*
r
*
in the numerator and
*
r
*
^{
2
}
in the denominator, we find that

Now find ∂
*
w
*
/ ∂
*
θ
*
by applying the Chain Rule for two independent variables.

Now we find ∂
*
w
*
/ ∂
*
θ
*
by applying the Chain Rule for two independent variables:

Substitute the expressions for
*
x
*
and
*
y
*
.

Substituting the expressions for
*
x
*
and
*
y
*
, we have

Factoring out
*
r
*
in the numerator and
*
r
*
^{
2
}
in the denominator yields