How do you use chain rule to find partial derivatives?

1 Answer
Bill K.
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.

Related questions
See all questions in Chain Rule
Impact of this question
3944 views around the world
You can reuse this answer
Creative Commons License