Apply the rules of partial differentiation on the following a. z = 4x2 – 8xy + 7y 5 – 3. Find dz/dx and dz/dy

To find the partial derivatives of the function z = 4x^2 - 8xy + 7y^5 - 3, we differentiate with respect to each variable in turn, treating all other variables as constants.

1. Partial derivative with respect to x (dz/dx):

The derivative of 4x^2 with respect to x is 8x. The derivative of -8xy with respect to x is -8y (because y is treated as a constant). The derivative of 7y^5 with respect to x is 0 (because it does not contain x). The derivative of -3 with respect to x is also 0 (because it is a constant).

So, dz/dx = 8x - 8y.

2. Partial derivative with respect to y (dz/dy):

The derivative of 4x^2 with respect to y is 0 (because it does not contain y). The derivative of -8xy with respect to y is -8x (because x is treated as a constant). The derivative of 7y^5 with respect to y is 35y^4 (using the power rule for differentiation). The derivative of -3 with respect to y is 0 (because it is a constant).

So, dz/dy = -8x + 35y^4.