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### Calculus Help

 The super chain rule AKA recursive chain rule for higher order compositions of functions.  We know the chain rule for a composite of two functions:d(fg) = f'(g(x))g'(x) dxBut what about the composite of more than two functions?  How would one differentiate (sin(x^2+1))^3  ?Here's a formula for repeated applications of the chain rule that I have never learned or seen formulated in any calculus classes anywhere.  Not claiming that I'm the first person to discover it, but I certainly am not borrowing this from elsewhere.d(fgh) = f'(g(h(x)))g'(h(x))h'(x)  dx or more compactly, d(fgh) = f'gh g'h h'   dxThe notation of Leibniz can make it even clearer.df  = df   dg  dhdx    dg  dh  dxThis can be extended indefinitely, assuming the domains of our functions allow it:d(fghijklm)  =    f'ghijklm g'hijklm h'ijklm i'jklm j'klm k'lm l'm m'        dxNow you can differentiate arbitrarily complex functions.  Try it for tan(sin(x^2+1))^3)You should get sec^2[((sin(x^2+1))^3]  3sin(x^2+1)^2 cos(x^2+1) 2x