Not Understanding This Derivation

[SweepingRocks]

I know this might be going back a few years in math lessons, but please bear with me.

https://imgur.com/jA97vSz <-- Here's the derivation I'm having issues with

So I understand the principles of why I'm supposed to be doing these steps, but the issue for me is going from that second line to the third line. This is the chain rule, I know, but when taking the derivative of the portion in the exponent, would the results not be -B/ln(c) (c^x) (c^t -1) (ln(c)) or -B/ln(c) (c^x) (c^t) (ln(c)) + B/ln(c) (c^x) ln(c) ?

When they derive it in the problem, it seems they are changing the ct - 1 from the second line to ct and I'm not seeing why. Perhaps if someone could do that part out or explain it, I'd get a better understanding.

[SweepingRocks]

It's because we're taking the derivative of t, not the derivative of x. How come I always figure these out right after I post them? :P

[Transitive Tangerine]

you're differentiating with respect to t, so treat everything else like a giant constant, so rewrite the exponent as: K(c^t - 1)

= Kc^t - K

If you take the derivative of that, you just get:
K(c^t)(lnc) - 0
= K(c^t)(lnc)

[Transitive Tangerine]

Originally Posted by SweepingRocks

It's because we're taking the derivative of t, not the derivative of x. How come I always figure these out right after I post them? :P

xD