Math Problem Statement

Tentukan ∂y/∂x, ∂y/∂z, ∂²y/∂x², ∂²y/∂z², dan ∂²y/∂x∂z dari fungsi majemuk berikut untuk y sebagai fungsi dari x dan z.

Solution

To solve the problem, we need to calculate the following partial derivatives for each function yy in parts (a), (b), and (c):

  1. First-order partial derivatives:

    • ∂y∂x\frac{\partial y}{\partial x}
    • ∂y∂z\frac{\partial y}{\partial z}

  2. Second-order partial derivatives:

    • ∂2y∂x2\frac{\partial^2 y}{\partial x^2}
    • ∂2y∂z2\frac{\partial^2 y}{\partial z^2}
    • Mixed partial derivative ∂2y∂x∂z\frac{\partial^2 y}{\partial x \partial z}

Let's go through each part in turn.

Part (a) y=6+3x+16z+4x2+2z2y = 6 + 3x + 16z + 4x^2 + 2z^2

  1. First-order partial derivatives:

    • ∂y∂x=3+8x\frac{\partial y}{\partial x} = 3 + 8x
    • ∂y∂z=16+4z\frac{\partial y}{\partial z} = 16 + 4z

  2. Second-order partial derivatives:

    • ∂2y∂x2=8\frac{\partial^2 y}{\partial x^2} = 8
    • ∂2y∂z2=4\frac{\partial^2 y}{\partial z^2} = 4
    • ∂2y∂x∂z=0\frac{\partial^2 y}{\partial x \partial z} = 0

Part (b) y=14x3z2y = 14x^3 z^2

  1. First-order partial derivatives:

    • ∂y∂x=42x2z2\frac{\partial y}{\partial x} = 42x^2 z^2
    • ∂y∂z=28x3z\frac{\partial y}{\partial z} = 28x^3 z

  2. Second-order partial derivatives:

    • ∂2y∂x2=84xz2\frac{\partial^2 y}{\partial x^2} = 84x z^2
    • ∂2y∂z2=28x3\frac{\partial^2 y}{\partial z^2} = 28x^3
    • ∂2y∂x∂z=84x2z\frac{\partial^2 y}{\partial x \partial z} = 84x^2 z

Part (c) y=9+4xz−3x−2z3y = 9 + 4xz - 3x^{-2}z^3

  1. First-order partial derivatives:

    • ∂y∂x=4z+6x−3z3\frac{\partial y}{\partial x} = 4z + 6x^{-3}z^3
    • ∂y∂z=4x−9x−2z2\frac{\partial y}{\partial z} = 4x - 9x^{-2}z^2

  2. Second-order partial derivatives:

    • ∂2y∂x2=−18x−4z3\frac{\partial^2 y}{\partial x^2} = -18x^{-4}z^3
    • ∂2y∂z2=−18x−2z\frac{\partial^2 y}{\partial z^2} = -18x^{-2}z
    • ∂2y∂x∂z=4−9x−3z2\frac{\partial^2 y}{\partial x \partial z} = 4 - 9x^{-3}z^2

Would you like further details on any specific part or derivative?

Here are some related questions that might deepen your understanding:

  1. How would the results change if we had different exponents on xx and zz in each function?
  2. What rules apply when calculating mixed partial derivatives for functions of multiple variables?
  3. Can the second mixed derivative ∂2y∂x∂z\frac{\partial^2 y}{\partial x \partial z} be different from ∂2y∂z∂x\frac{\partial^2 y}{\partial z \partial x}?
  4. How does the chain rule apply to functions of more than one variable?
  5. How would the derivatives look if an additional variable, say ww, was introduced?

Tip: When calculating partial derivatives, treat all variables except the one you're differentiating with respect to as constants.

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Math Problem Analysis

Mathematical Concepts

Partial Derivatives
Multivariable Calculus

Formulas

∂y/∂x - First-order partial derivative of y with respect to x
∂y/∂z - First-order partial derivative of y with respect to z
∂²y/∂x² - Second-order partial derivative of y with respect to x
∂²y/∂z² - Second-order partial derivative of y with respect to z
∂²y/∂x∂z - Mixed partial derivative of y with respect to x and z

Theorems

Schwarz's Theorem (Clairaut's Theorem on Equality of Mixed Partials)

Suitable Grade Level

University Calculus

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