Answer
(a) $\nabla{f} = 2x\textbf{x} + 3y^2\textbf{y} + 4z^3\textbf{z}$
(b) $\nabla{f} = 2xy^3z^4\textbf{x} + 3x^2y^2z^4\textbf{y} + 4x^2y^3z^3\textbf{z}$
(c) $\nabla{f} = e^xsin(y)ln(z)\textbf{x} + e^xcos(y)ln(z)\textbf{y} + e^xsin(y)(1/z)\textbf{z}$
Work Step by Step
For each part, one must carefully differentiate using the chain rule. For the first part, this is:
(a) $\frac{\partial f(x,y,z)}{\partial x} + \frac{\partial f(x,y,z)}{\partial y} + \frac{\partial f(x,y,z)}{\partial z} = \nabla{f}$ For which each term in $f(x,y,z)$ may be differentiated independently of other dependencies, as $x^2$, $y^3$, and $z^4$ are independent of each other.
(b) Since $x, y, z$ are all coupled, one must apply the product rule.
(c) This case is precisely similar to (b).
