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 needs the chain rule layered on the standard ex derivative (since the exponent is 2x, not x alone); the log term uses the standard lnx derivative directly.
Differentiating e2x as simply e2x (forgetting to multiply by the derivative of the exponent, 2, via the chain rule) — the standard-function result $\frac{d}{dx}(e^x …