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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 a single variable,
*
t
*
.

*
x
*
and
*
y
*
are functions of a single variable,
*
t
*
, so we find
*
dx
*
/
*
dt
*
and
*
dy
*
/
*
dt
*
:

Find
*
dw
*
/
*
dt
*
by applying the Chain Rule for one independent variable.

We find
*
dw
*
/
*
dt
*
by applying the Chain Rule for one independent variable:

Where do we go from here?

Substitute the expressions for
*
x
*
and
*
y
*
(in terms of
*
t
*
), and then differentiate (with respect to
*
t
*
).

Notice that the variable
*
t
*
does not appear in the expression for
*
dw
*
/
*
dt
*
, so treat
*
dw
*
/
*
dt
*
as a constant, and get
*
d
*
^{
2
}
*
w
*
/
*
dt
*
^{
2
}
= 0.

Express
*
dw
*
/
*
dt
*
in terms of
*
x
*
and
*
y
*
, and then go through the same process to find
*
d
*
^{
2
}
*
w
*
/
*
dt
*
^{
2
}
as we used to find
*
dw
*
/
*
dt
*
.

Replacing sin
*
t
*
with
*
y
*
, and cos
*
t
*
with
*
x
*
, in the expression for
*
dw
*
/
*
dt
*
, we get

*
dw
*
/
*
dt
*
is a function of two variables.

Since
*
dw
*
/
*
dt
*
is a function of two variables, we find its first partial derivatives:

Find
*
d
*
^{
2
}
*
w
*
/
*
dt
*
^{
2
}
by applying the Chain Rule for one independent variable to
*
dw
*
/
*
dt
*
.

We find
*
d
*
^{
2
}
*
w
*
/
*
dt
*
^{
2
}
by applying the Chain Rule for one independent variable to
*
dw
*
/
*
dt
*
(and using the expressions already found for
*
dx
*
/
*
dt
*
and
*
dy
*
/
*
dt
*
):

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

Substituting the expressions for
*
x
*
and
*
y
*
, we find that

Evaluate
*
d
*
^{
2
}
*
w
*
/
*
dt
*
^{
2
}
at the given value of
*
t
*
.

Substituting the given value of
*
t
*
, we find that at
*
t
*
= 0,