HallsofIvy said:
Partial derivatives involve finding the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant. In this case, the function is f(x-z)=x+y+z and we are finding the partial derivatives with respect to x and z.
To solve for the partial derivative with respect to x, you first treat z as a constant and differentiate the function as you would a single variable function. This means the derivative of x is 1, and the derivative of y and z are both 0. The final result would be 1.
To solve for the partial derivative with respect to z, you again treat x as a constant and differentiate the function. This time, the derivative of x is 0, the derivative of y is also 0, and the derivative of z is 1. The final result would be 1.
Yes, let's say we have the function f(x-z)=x+y+z and we want to find the partial derivative with respect to x at the point (2,3). We would plug in 2 for x and 3 for y and z, giving us f(2-3)=2+3+3=8. To find the partial derivative, we would use the formula f'(x) = lim(h->0) (f(x+h)-f(x))/h and plug in our values. This gives us f'(x) = (f(2+h-3)-f(2-3))/h = (8-8)/h = 0, so the partial derivative with respect to x at (2,3) is 0.
Partial derivatives are commonly used in fields such as physics, economics, and engineering to model and analyze complex systems. For example, in economics, partial derivatives can be used to find the marginal cost or marginal revenue of a product, helping businesses make decisions about pricing and production. In physics, partial derivatives are used to find rates of change in multivariable systems, such as the velocity of an object in 3D space. Overall, understanding how to solve partial derivatives can provide valuable insights and tools for problem-solving in many different fields.