Answer
a) ∂f /∂x|(1,1,1) = e
b) ∂f /∂y|(1,1,1) = 2e
c) ∂f /∂z|(1,1,1) = e
Work Step by Step
let f(x,y,z) = y^{2}e^{xz}
a) Calculate the partial derivative ∂f/∂x
∂f/∂x = ∂/∂x[y^{2}e^{xz}]
∂f/∂x = y^{2}∂/∂x[e^{xz}]
∂f/∂x = y^{2}ze^{xz}
Calculate the partial derivative ∂f/∂x at point (1,1,1)
∂f/∂x = y^{2}ze^{xz}
∂f /∂x|(1,1,1) = (1)^{2}(1)e^{1\times1}
∂f /∂x|(1,1,1) = e
b) Calculate the partial derivative ∂f /∂y
∂f /∂y = ∂/∂y[y^{2}e^{xz}]
∂f /∂y = e^{xz}d/dy[y^{2}]
∂f /∂y = 2ye^{xz}
Calculate the partial derivative ∂f/∂y at point (1,1,1)
∂f /∂y = 2ye^{xz}
∂f /∂y|(1,1,1) = 2(1)e^{1\times1}
∂f /∂y|(1,1,1) = 2e
c) Calculate the partial derivative ∂f/∂z
∂f/∂z = ∂/∂z[y^{2}e^{xz}]
∂f/∂z = y^{2}∂/∂z[e^{xz}]
∂f/∂z = y^{2}xe^{xz}
Calculate the partial derivative ∂f/∂z at point (1,1,1)
∂f/∂z = y^{2}xe^{xz}
∂f /∂z|(1,1,1) = (1)^{2}(1)e^{1\times1}
∂f /∂z|(1,1,1) = e
