Problem 42 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the following functions. $$g(x, y, z)=2 x^{2} y-3 x z^{4}+10 y^{2} z^{2}$$

Step-by-Step Solution

Answer

Question: Find the first partial derivatives of the function $$g(x, y, z) = 2x^2y - 3xz^4 + 10y^2z^2$$ with respect to x, y, and z. Answer: The first partial derivatives of the function are: - $$\frac{\partial g}{\partial x} = 4xy - 3z^4$$ - $$\frac{\partial g}{\partial y} = 2x^2 - 20yz^2$$ - $$\frac{\partial g}{\partial z} = -12xz^3 + 20y^2z$$

Step 1: Find the Partial Derivative with respect to x

To find the partial derivative of $$g(x, y, z)$$ with respect to x, we differentiate $$g(x, y, z)$$ with respect to x, treating y and z as constants. Using the rules of differentiation: $$\frac{\partial g}{\partial x} = \frac{\partial}{\partial x} (2x^2y - 3xz^4 + 10y^2z^2)$$ $$\frac{\partial g}{\partial x} = (2 * 2x * y - 3z^4 + 0)$$ $$\frac{\partial g}{\partial x} = 4xy - 3z^4$$

Step 2: Find the Partial Derivative with respect to y

To find the partial derivative of $$g(x, y, z)$$ with respect to y, we differentiate $$g(x, y, z)$$ with respect to y, treating x and z as constants. Using the rules of differentiation: $$\frac{\partial g}{\partial y} = \frac{\partial}{\partial y} (2x^2y - 3xz^4 + 10y^2z^2)$$ $$\frac{\partial g}{\partial y} = (0 + 2x^2 - 20yz^2)$$ $$\frac{\partial g}{\partial y} = 2x^2 - 20yz^2$$

Step 3: Find the Partial Derivative with respect to z

To find the partial derivative of $$g(x, y, z)$$ with respect to z, we differentiate $$g(x, y, z)$$ with respect to z, treating x and y as constants. Using the rules of differentiation: $$\frac{\partial g}{\partial z} = \frac{\partial}{\partial z} (2x^2y - 3xz^4 + 10y^2z^2)$$ $$\frac{\partial g}{\partial z} = (0 - 12xz^3 + 20y^2z)$$ $$\frac{\partial g}{\partial z} = -12xz^3 + 20y^2z$$ Now, we have found the first partial derivatives with respect to x, y, and z: - $$\frac{\partial g}{\partial x} = 4xy - 3z^4$$ - $$\frac{\partial g}{\partial y} = 2x^2 - 20yz^2$$ - $$\frac{\partial g}{\partial z} = -12xz^3 + 20y^2z$$