Chain rule for partial derivatives of multivariable functions (KristaKingMath)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange askers cite it as the source of the tree-diagram method for multivariable chain rule problems, and as a reference when confused about partial versus total derivatives.
Recommended by 1 people across 1 different months, 2013 to 2021.
What it is
A 15-minute calculus lesson by Krista King (KristaKingMath, also credited as integralCALC) on applying the chain rule to multivariable functions. It uses tree diagrams to organize partial derivatives when variables depend on other variables, with worked examples.
What people said
Recommends user759380 · 2020-03 · Mathematics Stack Exchange · on “Computing second order partial derivatives”
I am tasked with the following question: Consider the wave equation for linearly $x$ polarised waves travelling in the $\pm z$ directions: $$\frac{\partial^2 E_x}{\partial t^2}=c^2\frac{\partial^2 E_x}{\partial z^2}\tag{1}$$ Transform Eq. $(1)$ to the independent variables $q = z − ct$ and $s = z + ct$ and show that $$\frac{\partial^2 E_x}{\partial s \partial q}=0$$ When I am confronted with questions of this type I always use the 'tree diagram for partial derivatives' method which is outlined in this video - Chain rule and tree diagrams of multivariable functions (KristaKingMath) [this video] by Krista King. So writing $$E_{x}=E_{x}(q,s)$$ $$q=q(z,t) \qquad\text{and}\qquad s=s(z,t)$$ then $$\fra
Cites hertzsprung · 2013-10 · Mathematics Stack Exchange · on “Geometric intuition for partial derivatives with a single dependent variable”
Having watched an integralCALC video lesson [this video] , given $$w=xe^{y/z}, x = t^2, y = 1-t, z=1+2t$$ which could be rewritten as $$w=t^2 e^\frac{1-t}{1+2t}$$ How does $dw/dt$ differ from $\partial w/\partial t$? Is there some intuition behind the partial derivative (such as a geometric interpretation)?
Cites user395550 · 2021-05 · Mathematics Stack Exchange · on “When to write partial derivatives or total derivatives in the chain rule for multivariable functions?”
From the first page of this pdf on measurements of bandstructures [LINK] it is written that: [LINK] In the chain rule above, written again below for clarity: $$\frac{\partial k_x}{\partial t}\frac{\partial E}{\partial k_x}+\frac{\partial k_y}{\partial t}\frac{\partial E}{\partial k_y}+\frac{\partial k_z}{\partial t}\frac{\partial E}{\partial k_z}\equiv\frac{dE}{dt}$$ I am questioning whether the $\frac{dE}{dt}$ should actually be $\frac{\partial E}{\partial t}$ The reason I say this is because $E\equiv E({\bf{k}}) \equiv E(k_x, k_y, k_z)$ and $k_{x,y,z}\equiv k_{x,y,z}(t)$ I'm using the mnemonic method of tree diagrams outlined in the popular Youtube video by Krista King [this video] which for my