Find the gradients of the following functions:(a) f (x, y, z) = x2 + y3 + z4

The gradient of a function gives you the direction of the steepest ascent. The gradient is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is the rate of increase in that direction.

The gradient of a scalar function f(x, y, z) in three dimensions is given by the vector:

grad f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

where i, j, and k are the unit vectors in the x, y, and z directions respectively.

So, for the function f(x, y, z) = x^2 + y^3 + z^4, we can find the gradient by taking the partial derivatives with respect to x, y, and z.

∂f/∂x = 2x ∂f/∂y = 3y^2 ∂f/∂z = 4z^3

So, the gradient of f is:

grad f = 2xi + 3y^2j + 4z^3k