Hi, in my game, I use equations to determine where something is at a certain time, I use desmos.com to visualize this.

I’m trying to create an equivalence of the derivatives at 0 of the x axis, so that the transition of the 2 equations (red runs first, then blue) are seamless.

I have 2 different time periods where the domain of my equations is 0 to 1000, both times. Unfortunately this split up the code in a way to make it easier in other ways, but harder for this question. This is not something I want to change at this time.

This is equation1:

-12.385 + 1.0028^(x+900)

Wolffram alpha says that the derivative is:

0.0346293+1.0028^x

Equation2:

149.9-.996^(x-1250)

Equation2 derivative

0.600837+.996^x

Graphed below:

It does not seem right to me that the orange (corresponding as the derivative to the red equation) is so high around 1000 on the x axis, but then the purple derivative line for the blue equation is so low around 0 of the x axis. In the game, it seems to me that the object moves a lot quicker during the start to the blue equation than toward the end of the red.

I think it has to do with derivatives and using the format of something^(x-somethingElse)

Edit: Here’s a pic closer to the 0,0

Looks like it’d be correct to subtract one from the derivative equation, not sure if that fixes other areas though

Any input is appreciated.