If z = x² + y², then ∂z/∂x is?
A
2x
B
2y
C
x
D
y
Analysis & Theory
∂/∂x (x² + y²) = 2x.
If z = x²y, then ∂²z/∂x² is?
A
2y
B
2x
C
y
D
0
Analysis & Theory
∂z/∂x = 2xy ⇒ ∂²z/∂x² = 2y.
If z = xy², then ∂²z/∂x∂y is?
A
2y
B
x
C
y²
D
2x
Analysis & Theory
∂z/∂x = y² ⇒ ∂/∂y = 2y.
If z = sin(xy), then ∂²z/∂x∂y is?
A
cos(xy) - xy sin(xy)
B
cos(xy) - x² sin(xy)
C
cos(xy) - y² sin(xy)
D
cos(xy)
Analysis & Theory
∂z/∂x = y cos(xy) ⇒ ∂/∂y = cos(xy) - x² sin(xy).
If z = e^(x+y), then ∂²z/∂x² is?
A
e^(x+y)
B
2e^(x+y)
C
xe^(x+y)
D
ye^(x+y)
Analysis & Theory
∂z/∂x = e^(x+y) ⇒ ∂²z/∂x² = e^(x+y).
If z = ln(x² + y²), then ∂²z/∂y² is?
A
(2x² - 2y²)/(x²+y²)²
B
(2y² - 2x²)/(x²+y²)²
C
2(x² - y²)/(x²+y²)²
D
(-2y²)/(x²+y²)²
Analysis & Theory
∂z/∂y = 2y/(x²+y²). Differentiating: ∂²z/∂y² = (2x² - 2y²)/(x²+y²)².
If z = x² + xy + y², then ∂²z/∂x∂y is?
A
1
B
2
C
x
D
y
Analysis & Theory
∂z/∂x = 2x + y ⇒ ∂/∂y = 1.
If z = x³ + y³, then ∂²z/∂x² is?
A
6x
B
3x²
C
0
D
6y
Analysis & Theory
∂z/∂x = 3x² ⇒ ∂²z/∂x² = 6x.
If z = e^(xy), then ∂²z/∂x∂y is?
A
(1+xy)e^(xy)
B
(1+x)e^(xy)
C
(1+y)e^(xy)
D
(x+y)e^(xy)
Analysis & Theory
∂z/∂x = y e^(xy) ⇒ ∂/∂y = e^(xy) + xy e^(xy) = (1+xy)e^(xy).
If z = x²y², then ∂²z/∂y² is?
A
2x²
B
4x²y
C
2y²
D
x²
Analysis & Theory
∂z/∂y = 2x²y ⇒ ∂²z/∂y² = 2x².