# How do you find the derivative of #e^ [2 tan(sqrt x)]#?

##### 1 Answer

You do it one step at a time, keeping in mind the **chain rule**.

Any time you take the derivative of a function that contains a nested function (otherwise known as a **composite function**), take the derivative of the nested function as well.

That is,

- The derivative of
#f(u) = e^u# is#e^u ((du)/(dx))# . - The derivative of
#f(u) = tanu# is#sec^2u ((du)/(dx))# - The derivative of
#f(x) = sqrtx# is#1/(2sqrtx)# .

Therefore:

#color(blue)(d/(dx)[e^(2tansqrtx)])#

#= e^(2tansqrtx) * stackrel("Chain Rule")overbrace(2d/(dx)[tansqrtx])#

Here,

#=> e^(2tansqrtx) * 2(sec^2sqrtx * stackrel("Chain Rule Again")overbrace(d/(dx)[sqrtx]))#

#= cancel(2)e^(2tansqrtx)sec^2sqrtx * 1/(cancel(2)sqrtx)#

And here,

#=> color(blue)((e^(2tansqrtx)sec^2sqrtx)/sqrtx)#

That's as simple an answer as it gets, so don't be surprised if you get this. :-)