Write formulas for the indicated partial derivatives for the multivariable function.

Question

f(x,y)=3x^3+2x^2y^3+8xy+14x+7

a) ∂f / ∂x

b) ∂f / ∂y

c) ∂f / ∂x y=3

Answer 1

a) differentiate with respect to x while treating the other variable, y, like a constant:

∂/∂x( 3x^3) = 9x^2

∂/∂x(2x^2y^3) = y^3(∂/∂x(2x^2)) = y^3(4x) = 4xy^3

∂/∂x(8xy) = y(∂/∂x(8x)) = y(8) = 8y

∂/∂x(14x) = 14

∂/∂x(7) = 0

so the answer is:

a) is: ∂f / ∂x = 9x^2 + 4xy^3 + 8y + 14

b) differentiate with respect to y while treating the other variable, x, like a constant:

∂/∂y(3x^3) = 3x^3(∂/∂y(1)) = 3x^3(0) = 0

∂/∂y(2x^2y^3) = 2x^2(∂/∂x(y^3)) = 2x^2(3y^2) = 6x^2y^2

∂/∂y(8xy) = x(∂/∂x(8y)) = x(8) = 8x

∂/∂y(14x) = 14x(∂/∂y(1)) = 14x(0) = 0

∂/∂y(7) = 0

so the answer is:

b) is: ∂f / ∂y = 6x^2y^2 + 8x

c) plug y = 3 into ∂f/∂x given in a) and simplify



∂f / ∂x = 9x^2 + 4xy^3 + 8y + 14, so at y = 3 (just substitute 3 everywhere you see y and simplify)

∂f / ∂x_(y=3) = 9x^2 + 4x(3^3) + 8(3) + 14 = 9x^2 + 108x + 24 + 14 = 9x^2 + 108x +38

so the answer is:

c) ∂f / ∂x_(y=3) = 9x^2 + 108x +38