Q.Find the derivative of the following function with respect to the corresponding independent variable: $y = \sin x + \cos x$

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.

Logarithmic and exponential functions.

($e^x$ is the unique elementary function that is its own derivative — the special case $a=e$, $\log e=1$.)

The six trigonometric functions.

Only $\sin x\to\cos x$ is derived directly from the limit definition (via the sum-to-product identity and $\lim_{\theta\to0}\sin\theta/\theta=1$); every other trig derivative follows from it using the chain rule ($\cos x=\sin(\tfrac\pi2+x)$) or the quotient rule ($\tan x=\sin x/\cos x$, etc.) — so the whole trig table rests on a single limit.

The six inverse trigonometric functions (each domain-restricted to its principal branch):

$\sin^{-1}x$ and $\tan^{-1}x$ are derived by writing $x=\sin y$ (resp. $x=\tan y$) and differentiating implicitly (§10.4.3's technique, used here before that section is formally reached); the three complementary pairs ($\cos^{-1}\leftrightarrow\sin^{-1}$, $\cot^{-1}\leftrightarrow\tan^{-1}$, and $\operatorname{cosec}^{-1}\leftrightarrow\sec^{-1}$) each follow from a co-function identity such as $\sin^{-1}x+\cos^{-1}x=\tfrac\pi2$ (a constant, so the two derivatives are exact negatives of each other).