What is the derivative of $5^tanx$ ?

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Answer 1 · 2015-07-17T22:23:57

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.)

What is the derivative of 5^tanx5^tanx?