Can someone please walk me through the steps for the following Partial derivatives problems:
f(x,y) = x^7 - 8y - 3, find x and y
f(x,y) = 7 / 5x+4y, find x and y
To find the partial derivatives for the given functions, you need to use the chain rule and the power rule. I will walk you through the steps for each problem.
1. f(x, y) = x^7 - 8y - 3
To find the partial derivative with respect to x (∂f/∂x), treat y as a constant and differentiate only with respect to x. Similarly, to find the partial derivative with respect to y (∂f/∂y), treat x as a constant and differentiate only with respect to y.
∂f/∂x = d/dx(x^7) - d/dx(8y) - d/dx(3)
∂f/∂x = 7x^6 - 0 - 0
∂f/∂x = 7x^6
∂f/∂y = d/dy(x^7) - d/dy(8y) - d/dy(3)
∂f/∂x = 0 - 8 - 0
∂f/∂x = -8
2. f(x, y) = 7 / (5x + 4y)
Again, to find the partial derivative with respect to x (∂f/∂x) and y (∂f/∂y), we treat the other variable as a constant.
∂f/∂x = d/dx(7) - d/dx(5x + 4y)
∂f/∂x = 0 - 5/(5x + 4y)^2
∂f/∂y = d/dy(7) - d/dy(5x + 4y)
∂f/∂y = 0 - 4/(5x + 4y)^2
These are the partial derivatives of the given functions with respect to x and y.