Multivariate Calculus
IMPORTANT You will need to have a WolframAlpha Pro account to use Wolfram effectively.
Partial Derivatives
You can find partial derivatives in exactly the same way as you would use Wolfram for full derivatives: type d/dx or d/dy in front of your function:

Second-Order Partial Derivatives
2nd order partial derivatives can be done in much the same way. You can type d2/dxdy or d2/dx2 in front of your equation:

Integration
Integration with Wolfram works similarly to differentiation. Type ∫ from the Wolfram extended keyboard in front of your function, and dy, dx etc after it, like this.
