Concept understanding — Derivatives of Standard Functions
Applying the first-principle limit once to each basic elementary function builds a permanent table; every later problem then differentiates by combining this table with the rules of differentiation (sum/product/quotient/chain/constant-multiple), with no further limit ever required.
Algebraic functions.
dxd(k)=0(k constant),dxd(xn)=nxn−1for any real n(Corollaries 10.1–10.2 extend the integer case to rational, then any real, exponent).
Only sinx→cosx is derived directly from the limit definition (via the sum-to-product identity and limθ→0sinθ/θ=1); every other trig derivative follows from it using the chain rule (cosx=sin(2π+x)) or the quotient rule (tanx=sinx/cosx, etc.) — so the whole trig table rests on a single limit.
The six inverse trigonometric functions (each domain-restricted to its principal branch):
The exponential term uses the standard ex derivative directly (the exponent is plain x); the log term needs the chain rule since the argument is 3x, not x alone.