Multivariable function: $x^3 + y^3 + e^{zw}$ - what to do after taking partial derivatives?

Question

I'm working on this problem. I'm kinda stuck and would be very grateful for any help. A have to compute the gradient of this function

$$f(x,y,z,w) = x^3 + y^3 + e^{zw}$$

So, from what I know I should take the partial derivative with respect to each variable:

$$\begin{align} \frac{\mathrm df}{\mathrm dx} &= 3x \\ \frac{\mathrm df}{\mathrm dy} &= 3y \\ \frac{\mathrm df}{\mathrm dz} &= we^z \\ \frac{\mathrm df}{\mathrm dw} &= ze^w \end{align}$$

My problem is, is that the right approach? what do I do from here?

Answer 1

$$ \nabla f \stackrel{\text{def}}{=} \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}, \frac{\partial f}{\partial w} \right).$$

Answer 2

You are right that you should take the partial derivative with respect to each variable, but all your derivatives are wrong. The correct derivatives are: $$\frac{\partial f}{\partial x} = 3x^2$$ $$\frac{\partial f}{\partial y} = 3y^2$$ $$\frac{\partial f}{\partial z} = we^{wz}$$ $$\frac{\partial f}{\partial w} = ze^{wz}$$ The gradient is then the vector valued function $$\nabla f(x,y,z,w) = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \\ \frac{\partial f}{\partial w} \end{bmatrix} = \begin{bmatrix} 3x^2 \\ 3y^2 \\ we^{wz} \\ ze^{wz} \end{bmatrix}$$