Find the gradients of the following functions:(a) f (x, y, z) = x2 + y3 + z4 .(b) f (x, y, z) = x2 y3 z4 .(c) f (x, y, z) = e x sin(y) ln(z)

Sure, let's find the gradients of the given functions.

(a) f (x, y, z) = x^2 + y^3 + z^4

The gradient of a function is a vector of its partial derivatives. So, we need to find the partial derivatives of f with respect to x, y, and z.

∂f/∂x = 2x ∂f/∂y = 3y^2 ∂f/∂z = 4z^3

So, the gradient of f is (2x, 3y^2, 4z^3).

(b) f (x, y, z) = x^2 y^3 z^4

Again, we find the partial derivatives:

∂f/∂x = 2xy^3z^4 ∂f/∂y = 3x^2y^2z^4 ∂f/∂z = 4x^2y^3z^3

So, the gradient of f is (2xy^3z^4, 3x^2y^2z^4, 4x^2y^3z^3).

(c) f (x, y, z) = e^x sin(y) ln(z)

The partial derivatives are:

∂f/∂x = e^x*sin(y)ln(z) ∂f/∂y = e^xcos(y)ln(z) ∂f/∂z = e^xsin(y)/z

So, the gradient of f is (e^x*sin(y)ln(z), e^xcos(y)ln(z), e^xsin(y)/z).