Remembering the following formulas has been a nuisance for me for years now.
Common Derivatives
Common Integrals
Can anyone suggest the best way to remember them?
Here is a trick I use to remember the derivatives and antiderivatives of trigonometric functions. If you know that \begin{align} \sin'(x) &= \cos (x) \\ \sec'(x) &= \sec (x)\tan(x) \\ \tan'(x) &= \sec^2(x) \, , \end{align} then the derivatives of $\cos$, $\cot$, and $\csc$ can be memorised with not much more effort. These functions have the prefix co- in them for a reason: cosine is the sine of the co-angle, cotangent is the tangent of the co-angle, and cosecant is the secant of the co-angle. This leads to the identities \begin{align} \cos x &= \sin\left(\frac{\pi}{2}-x\right) \\[4pt] \csc x &= \sec\left(\frac{\pi}{2}-x\right) \\[4pt] \cot x &= \tan\left(\frac{\pi}{2}-x\right) \, . \end{align} These identities, along with the chain rule, can be used to find the derivatives of the “co-functions": \begin{align} \frac{d}{dx}\bigl(\cos x\bigr) &= \frac{d}{dx}\bigl(\sin(\pi/2-x)\bigr)=-\cos(\pi/2-x)=-\sin x \\[4pt] \frac{d}{dx}\bigl(\csc x\bigr) &= \frac{d}{dx}\bigl(\sec(\pi/2 - x) \bigr) = -\sec(\pi/2 - x)\tan(\pi/2 - x) = -\csc x \cot x \\[4pt] \frac{d}{dx}\bigl(\cot x\bigr) &= \frac{d}{dx}\bigl(\tan(\pi/2 - x)\bigr)= -\sec^2(\pi/2 - x) = -\csc^2 x \, . \end{align} In general, if $$ \frac{d}{dx}\left(\operatorname{something}(x)\right)=\operatorname{anotherthing}(x) $$ then $$ \frac{d}{dx}\left(\operatorname{cosomething}(x)\right)=-\operatorname{coanotherthing}(x) \, . $$ This trick also works for integrals: $$ \int \tan x = -\ln(\cos x)+C $$ and so $$ \int \cot x = \ln(\sin x) + C \, . $$
For inverse functions, use the formula $$ (f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))} \, . $$ The lesser-known $$ \int f^{-1}(x) \, dx = xf^{-1}(x) - F(f^{-1}(x)) + C \, , $$ where $F$ is an antiderivative of $f$, also comes in handy. This formula can be derived using integration by parts.
Memorize the derivatives of $x^n$, $e^x$, $\ln|x|$, $\sin x$, $\cos x$, $\arcsin x$, $\arctan x$, and maybe $\tan x$ (which are used all the time), and derive the rest whenever you need them (which isn't often, in my experience).
Then many of the integrals will just be “backwards versions” of what you already know, so there's no extra memory required to store them, and for those that are not, you can compute them when needed rather than memorizing them. For example, $$ \int \ln x \, dx = \int 1 \cdot \ln x \, dx = \cdots \quad \text{(integrate by parts)} $$ or $$ \int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx = - \int \frac{-\sin x}{\cos x} \, dx = - \ln|\cos x| + C \qquad \text{(pattern recognition, $\tfrac{f'(x)}{f(x)}$)} . $$ One tricky case, which I would recommend memorizing (even though it's not included in your list) is $$ \int \frac{dx}{\sqrt{a^2+x^2}} = \ln\left|x + \sqrt{a^2+x^2}\right| + C . $$
The way I always remembered derivatives of sine and cosine, secant, etc. is by remembering this one chain:
$$s\to c\to -s\to -c.$$ Each arrow represents differentiation, and $s$ is for sine and $c$ for cosine. From here, I also remember that $$\frac{1}{\sin u}=\csc u, \frac{1}{\cos u}=\sec u.$$ That reciprocals start with the opposite letter: that is, the reciprocal of sine which starts with "s" is cosecant which starts with "c". Similarly for cosine which starts with "c" whose reciprocal is secant which starts with an "s".
After this, I remember that
$$\tan u=\frac{\sin u}{\cos u}\iff D_u\tan u=\frac{\cos^2u-\sin^2u}{\cos^2u}=1-\tan^2u.$$ All these, coupled with the relation $$\cos^2u+\sin^2u=1,$$ and using the differential relations to obtain integral relations, the majority of my cosine sine memories are above, with the exception of these extras:
$$\sin(u+v)=\sin u\cos v+\cos u\sin v,\\ D^n\begin{matrix}\cos \\ \sin \end{matrix}u=\begin{matrix}\cos \\ \sin \end{matrix}\left(u+\frac{\pi n}{2}\right)$$
