If z = x²y + y³, then ∂z/∂x is?
A
2xy
B
y²
C
x² + 3y²
D
2x + y³
Analysis & Theory
Treat y as constant: ∂/∂x (x²y + y³) = 2xy.
If z = e^(xy), then ∂z/∂x is?
A
ye^(xy)
B
xe^(xy)
C
e^(xy)
D
(x+y)e^(xy)
Analysis & Theory
∂/∂x (e^(xy)) = y·e^(xy).
If z = ln(x² + y²), then ∂z/∂y is?
A
2y/(x² + y²)
B
2x/(x² + y²)
C
1/(x² + y²)
D
y/(x² + y²)
Analysis & Theory
∂/∂y [ln(x² + y²)] = (1/(x²+y²))·2y.
If z = x³ + y³, then ∂²z/∂x² is?
A
3x²
B
6x
C
0
D
3y²
Analysis & Theory
∂z/∂x = 3x² ⇒ ∂²z/∂x² = 6x.
If z = sin(xy), then ∂z/∂y is?
A
x cos(xy)
B
y cos(xy)
C
cos(xy)
D
sin(x)
Analysis & Theory
∂/∂y [sin(xy)] = cos(xy)·x.
If z = x² + xy + y², then ∂z/∂x is?
A
2x + y
B
2y + x
C
x + y
D
2x + 2y
Analysis & Theory
∂/∂x (x² + xy + y²) = 2x + y.
If z = x²y², then ∂²z/∂x∂y is?
A
2xy
B
4xy
C
x²
D
y²
Analysis & Theory
∂z/∂x = 2xy² ⇒ ∂/∂y = 4xy.
If z = e^(x² + y²), then ∂²z/∂x² is?
A
(4x²+2)e^(x²+y²)
B
2xe^(x²+y²)
C
4x²e^(x²+y²)
D
(2x)e^(x²+y²)
Analysis & Theory
∂z/∂x = 2x e^(x²+y²). Differentiating again: ∂²z/∂x² = (4x²+2)e^(x²+y²).
If z = xy + cos(y), then ∂z/∂y is?
A
x - sin(y)
B
x + sin(y)
C
y + cos(y)
D
x - cos(y)
Analysis & Theory
∂/∂y (xy + cos(y)) = x - sin(y).
If z = ln(xy), then ∂²z/∂x∂y is?
A
1/(xy)
B
0
C
-1/(x²y²)
D
-1/(xy²)
Analysis & Theory
∂z/∂x = 1/x ⇒ ∂/∂y (1/x) = 0, but using product form ln(xy) ⇒ ∂²z/∂x∂y = 1/(xy).