Question

What is the derivative of `5^tanx`?

Answer 1

Let's derive what the derivative is, because I don't remember what it is (yet).

`log_5(y) = log_5(5^tanx) = tanx`

Change of Base law:
`(lny)/(ln5) = tanx`

`lny = tanx ln5`

Implicit Differentiation:
`1/y((dy)/(dx)) = ln5 * sec^2x`

`(dy)/(dx) = y[ln5 * sec^2x]`

`= color(blue)(5^(tanx)[ln5*sec^2x])`

Yep, there it is. So then the general derivative is:

`color(green)(d/(dx)[b^u] = b^u*lnb*(du)/(dx))`

(see how you can just figure it out without remembering it? It's a trick.)