Efficient Calculation of Partial Derivatives ∂f/∂x and ∂f/∂y

For ∂f/∂x:
∂/∂x(2x + 2y)	= Start with the original function
= ∂/∂x(2x) + ∂/∂x(2y)	= Apply the sum rule for differentiation
= 2 ∂/∂x(x) + 0	= Partial differentiate each term
= 2 × 1 + 0	= Evaluate the partial derivatives
= 2	= Simplify
For ∂f/∂y:
∂/∂y(2x + 2y)	= Start with the original function
= ∂/∂y(2x) + ∂/∂y(2y)	= Apply the sum rule for differentiation
= 0 + 2 ∂/∂y(y)	= Partial differentiate each term
= 0 + 2 × 1	= Evaluate the partial derivatives
= 2	= Simplify

Detailed Calculation of Partial Derivatives ∂f/∂x and ∂f/∂y for f(x, y) = 4x² + 2y

For ∂f/∂x:
∂/∂x(4x² + 2y)	= Start with the original function
= ∂/∂x(4x²) + ∂/∂x(2y)	= Apply the sum rule for differentiation
= 2 * 4 * x¹	= Apply the power rule on 4x², knowing 2y is constant with respect to x
= 8 * x	= Simplify, 2 * 4 = 8
= 8x	= Write in simplified form
For ∂f/∂y:
∂/∂y(4x² + 2y)	= Start with the original function
= 0 + ∂/∂y(2y)	= 4x² is constant with respect to y
= 0 + 2	= Differentiate 2y with respect to y
= 2	= Write in simplified form

Detailed Calculation of Partial Derivatives ∂f/∂x and ∂f/∂y for \( f(x, y) = 4x² + 2y³ + 4 \)

For ∂f/∂x:
∂/∂x(4x² + 2y³ + 4)	= Start with the original function
= ∂/∂x(4x²) + ∂/∂x(2y³) + ∂/∂x(4)	= Apply the sum rule for differentiation
= 2 * 4 * x¹ + 0 + 0	= Apply the power rule on 4x²; 2y³ and 4 are constants with respect to x
= 8 * x	= Simplify, 2 * 4 = 8
= 8x	= Write in simplified form
For ∂f/∂y:
∂/∂y(4x² + 2y³ + 4)	= Start with the original function
= 0 + ∂/∂y(2y³) + 0	= 4x² and 4 are constants with respect to y
= 0 + 3 * 2 * y²	= Differentiate 2y³ with respect to y
= 6 * y²	= Simplify, 3 * 2 = 6
= 6y²	= Write in simplified form

∂/∂x(x³y – x + 4)	Start by writing down the function f(x, y) and indicating we want the partial derivative with respect to x.
= ∂/∂x(x³y) + ∂/∂x(-x) + ∂/∂x(4)	Apply the linearity property of derivatives to break down into simpler parts.
= y * 3x² – 1	Perform the differentiation: the derivative of x³y with respect to x is y * 3x², and the derivative of -x is -1.
= 3x²y – 1	Simplify the expression to get the final result for ∂f/∂x.

∂/∂y(x³y – x + 4)	Start by writing down the function f(x, y) and indicating we want the partial derivative with respect to y.
= ∂/∂y(x³y)	Since x and 4 are constants with respect to y, their derivatives become zero, leaving us with only ∂/∂y(x³y).
= x³	Perform the differentiation: the derivative of x³y with respect to y is simply x³.

∂(x/y)/∂x	Start by indicating we want to find the partial derivative of f(x, y) = x/y with respect to x.
∂(x * y⁻¹)/∂x	Express x/y as x * y⁻¹ for easier differentiation.
y⁻¹	Perform the differentiation. The derivative of x with respect to x is 1. Multiply by y⁻¹ to get y⁻¹.
1/y	Simplify y⁻¹ to 1/y, achieving the final result for ∂f/∂x.

∂(x/y)/∂y	Start by indicating we want to find the partial derivative of f(x, y) = x/y with respect to y.
∂(x * y⁻¹)/∂y	Express x/y as x * y⁻¹ for easier differentiation.
-x * y⁻²	Perform the differentiation. The derivative of y⁻¹ with respect to y is -y⁻². Multiply by x to get -x * y⁻².
-x/y²	Simplify -x * y⁻² to -x/y², achieving the final result for ∂f/∂y.

∂/∂x(xy)	Initial expression for partial derivative with respect to x.
= y ∂/∂x(x)	Apply the differentiation rules, treating y as a constant.
= y × 1	Derivative of x with respect to x is 1.
= y	Simplifying, we get y.

∂/∂y(xy)	Initial expression for partial derivative with respect to y.
= x ∂/∂y(y)	Apply the differentiation rules, treating x as a constant.
= x × 1	Derivative of y with respect to y is 1.
= x	Simplifying, we get x.