Question

How do you use chain rule to find partial derivatives?

Answer 1

One type of example is as follows: If `z=f(x,y)`, `x=g(t)`, and `y=h(t)`, then you can write `\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}`.

For example, if `z=x^2+y^3`, `x=t^4`, and `y=t^5`, then `\frac{\partial z}{\partial t}=2x\cdot 4t^3+3y^2\cdot 5t^{4}=8t^4\cdot t^3+15t^10\cdot t^4=8t^7+15t^14.`

You can check that this works by noting that `z=x^2+y^3=t^8+t^15` so that `8t^7+15t^14` (so the Chain Rule is more work for this example).

There are other examples where the Chain Rule can be less work.