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

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

My problem is, is that the right approach? what do I do from here?
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
$$\nabla f(x,y,z,w) = \begin{bmatrix}
           \frac{\partial f}{\partial x} \\
           \frac{\partial f}{\partial y} \\
           \frac{\partial f}{\partial z} \\
           \frac{\partial f}{\partial w}
         \end{bmatrix} = \begin{bmatrix}
           3x^2 \\
           3y^2 \\
           we^{wz} \\
           ze^{wz}
         \end{bmatrix}$$
$$ \nabla f \stackrel{\text{def}}{=} \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}, \frac{\partial f}{\partial w} \right).$$