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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,
*
s
*
and
*
t
*
.

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

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

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

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

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

Recall the double–angle identity for cos
*
t
*
.

Recalling the double–angle identity for cos
*
t
*
,

cos(2
*
t
*
) = (cos
^{
2
}
*
t
*
) – (sin
^{
2
}
*
t
*
),

we see that

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

Now we find ∂
*
w
*
/ ∂
*
t
*
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

Recall the double–angle identity for sin
*
t
*
.

Recalling the double–angle identity for sin
*
t
*
,

sin(2
*
t
*
) = 2(sin
*
t
*
)(cos
*
t
*
),

we see that

Evaluate the first partial derivatives of
*
w
*
at the given point.

Substituting the given data, we find that when
*
s
*
= 3 and
*
t
*
=
*
π
*
*
*
/4,