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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


Master the Concept of Partial Derivatives with f(x, y) = x³y – x + 4

Unlocking the Secrets of ∂f/∂x:

∂/∂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.

Demystifying the Partial Derivative ∂f/∂y:

∂/∂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.

Unlock the Complexity of Partial Derivatives with f(x, y) = x/y

Dive Into the Mysteries of ∂f/∂x:

∂(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.

Discover the Partial Derivative ∂f/∂y:

∂(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.

Dive Deep into Partial Derivatives: A Detailed Walkthrough of \( f(x, y) = xy \)

Understanding ∂f/∂x:

∂/∂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.

Understanding ∂f/∂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.

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