How do you use chain rule to find partial derivatives?

1 Answer
Apr 1, 2015

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.