magento 2 - Terms and conditions checkbox on register page

In the first one, if $v$ is a constant, then the chain rule is not involved. If $v$ is not a constant, but is a function of $u$, then your formula is wrong, and some form of the chain rule is needed. One way to go about it is $u^{v(u)}=e^{v(u)\log u}$; the derivative is $$e^{v(u)\log u}\times{d\over du}(v(u)\log u)=e^{v(u)\log u}\left({dv\over du}\log u+{v(u)\over u}\right)$$

Similar remarks apply to the second one, this time with the question being whether $u$ is a constant or a function of $v$.

Where to begin?

The most obvious problem you have is layout. Switching supplies require that the power and ground paths be as short and as thick as possible. That specifically includes the connection to IC2. Try to emulate the layout of the commercial PC board.

What is your "special supply"? It apparently is able to source more than 5 amps, if that's where your fuse current is coming from, but I doubt that it is.

Where, exactly, is your "shortstopper fuse"?

Assuming your fuse is on the battery output, the most likely problem is that your output voltage is too low, and the battery is back-driving the circuit. Put a high-current (5 amps or more) diode between the output and the battery, allowing only charge current to enter the battery. Before you connect the battery, put a 10 ohm resistor on the output, and verify that the load voltage is more than the battery no-load voltage.

The MOSFETs are rated for 20 to 30 amps continuous, with pulse rating over 100 amps, so as long as they are being properly driven you should have no worries there.

Why do you think a gate/source resistor on M3 will help? That's just idle curiosity on my part, but the fact that you think it will suggests that you need to learn more about how the circuit works. Although, to be fair, I'd really like to see the rationale for M3 being there in the first place.