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Partial Derivatives
bencejdanko
What is the notation for the partial derivative of
f
f
f
with respect to
x
x
x
?
Δ
f
x
\Delta f_x
Δ
f
x
∂
f
∂
x
\frac{\partial f}{\partial x}
∂
x
∂
f
d
f
d
x
\frac{df}{dx}
d
x
df
f
⋅
x
f \cdot x
f
⋅
x
Given
f
(
x
,
y
)
=
3
x
2
y
+
5
y
3
f(x, y) = 3x^2y + 5y^3
f
(
x
,
y
)
=
3
x
2
y
+
5
y
3
, find
∂
f
∂
x
\frac{\partial f}{\partial x}
∂
x
∂
f
.
6
x
6x
6
x
3
x
2
+
15
y
2
3x^2 + 15y^2
3
x
2
+
15
y
2
6
x
y
6xy
6
x
y
6
x
y
+
5
y
3
6xy + 5y^3
6
x
y
+
5
y
3
Given
f
(
x
,
y
)
=
3
x
2
y
+
5
y
3
f(x, y) = 3x^2y + 5y^3
f
(
x
,
y
)
=
3
x
2
y
+
5
y
3
, find
∂
f
∂
y
\frac{\partial f}{\partial y}
∂
y
∂
f
.
6
x
y
+
15
y
2
6xy + 15y^2
6
x
y
+
15
y
2
3
x
2
3x^2
3
x
2
15
y
2
15y^2
15
y
2
3
x
2
+
15
y
2
3x^2 + 15y^2
3
x
2
+
15
y
2
Compute the partial derivative
∂
∂
x
(
x
e
y
)
\frac{\partial}{\partial x}(x e^y)
∂
x
∂
(
x
e
y
)
.
0
0
0
e
x
e^x
e
x
x
e
y
x e^y
x
e
y
e
y
e^y
e
y
If
f
(
x
,
y
)
=
x
2
y
3
f(x, y) = x^2 y^3
f
(
x
,
y
)
=
x
2
y
3
, what is the mixed partial derivative
f
x
y
f_{xy}
f
x
y
(also written
∂
2
f
∂
y
∂
x
\frac{\partial^2 f}{\partial y \partial x}
∂
y
∂
x
∂
2
f
)?
12
x
y
12xy
12
x
y
6
x
2
y
6x^2y
6
x
2
y
6
x
y
2
6xy^2
6
x
y
2
2
x
+
3
y
2
2x + 3y^2
2
x
+
3
y
2
The vector composed of all first-order partial derivatives of a scalar function is called the:
Jacobian
Determinant
Gradient
Hessian
Geometrically, what does
∂
f
∂
x
\frac{\partial f}{\partial x}
∂
x
∂
f
represent at a point
(
a
,
b
)
(a,b)
(
a
,
b
)
on the surface
z
=
f
(
x
,
y
)
z=f(x,y)
z
=
f
(
x
,
y
)
?
The slope of the tangent line in the
x
x
x
-direction.
The volume under the surface.
The curvature of the surface.
The height of the surface at that point.
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