91Ó°ÊÓ

To find the partial derivative of \(f(x, y)\) and \(f(x, y, z)\) with respect to x, we treat y in the case of \(f(x, y)\) and y and z in the case of \(f(x, y, z)\) as constants and differentiate the function with respect to x. This will give us \(\frac{\partial f}{\partial x}\) for both functions.

Similar to Step 1, we now find the partial derivative of \(f(x, y)\) and \(f(x, y, z)\) with respect to y. Here, we treat x as a constant for both functions and additionally z as a constant for \(f(x, y, z)\). Then differentiate the function with respect to y. This will yield \(\frac{\partial f}{\partial y}\) for both functions.

For the three-variable function \(f(x, y, z)\), we need to find the partial derivative with respect to z. In this case, we treat x and y as constants, and differentiate the function with respect to z. This will give us \(\frac{\partial f}{\partial z}\).

For the two-variable function \(f(x, y)\), the gradient will be a vector with two components, where the first component is \(\frac{\partial f}{\partial x}\) and the second component is \(\frac{\partial f}{\partial y}\). So the gradient of \(f(x, y)\) is represented as: $$\nabla f(x, y) = \begin{bmatrix} \frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y} \end{bmatrix}$$ For the three-variable function \(f(x, y, z)\), the gradient will be a vector with three components, where the first component is \(\frac{\partial f}{\partial x}\), the second component is \(\frac{\partial f}{\partial y}\), and the third component is \(\frac{\partial f}{\partial z}\). The gradient of \(f(x, y, z)\) is represented as: $$ \nabla f(x, y, z) = \begin{bmatrix} \frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\\ \frac{\partial f}{\partial z} \end{bmatrix}$$