how to solve partial differentiation chain rule?

Sure, here's a step-by-step guide on how to solve partial differentiation using the chain rule:

1. Understand the Chain Rule: The chain rule is a formula to compute the derivative of a composite function. In the context of partial derivatives, the chain rule becomes a bit more complex because you have to take into account the rate of change in all variables.

2. Identify the Functions: Suppose you have a function h(x, y) = f(g(x, y), k(x, y)). Here, f is a function of two variables u and v, where u = g(x, y) and v = k(x, y).

3. Differentiate: The partial derivative of h with respect to x is given by: ∂h/∂x = (∂f/∂u * ∂u/∂x) + (∂f/∂v * ∂v/∂x). Similarly, the partial derivative of h with respect to y is given by: ∂h/∂y = (∂f/∂u * ∂u/∂y) + (∂f/∂v * ∂v/∂y).

4. Compute the Derivatives: Now, you just need to compute the derivatives on the right hand side of these equations. Remember, when you're computing ∂f/∂u, you treat v as a constant, and vice versa.

5. Substitute Back: Once you've computed the derivatives, substitute the functions u and v back into your equations to get the final answer.

Remember, practice is key when it comes to mastering the chain rule for partial derivatives. So, make sure to solve a lot of problems to get a good grip on this concept.