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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