What are the rules for finding partial derivatives of z=f(x,y)?
To find the partial derivatives of z = f(x,y), the following rules apply:
1. To find the partial derivative of f(x,y) with respect to x (denoted by ∂f/∂x), treat y as a constant and differentiate f(x,y) with respect to x.
2. To find the partial derivative of f(x,y) with respect to y (denoted by ∂f/∂y), treat x as a constant and differentiate f(x,y) with respect to y.
3. When differentiating terms with multiple variables, treat all variables except the one being differentiated with respect to as constants.
4. When differentiating products of variables, use the product rule: differentiate the first term with respect to the variable, then multiply by the second term and add to the product of the first term and the derivative of the second term with respect to the variable.
5. When differentiating compositions of functions, use the chain rule: differentiate the outer function with respect to the variable and multiply by the derivative of the inner function with respect to the variable.
6. If the partial derivatives are continuous in a region, then the function is said to be partially differentiable in that region.