Question

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

Answer 1

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, `d/(dx)[f(u(x))] = (df(u))/(du(x))*(du(x))/(dx)`. So:

• 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, `u = tansqrtx`, so `color(green)(d/(dx)[u(x)] = sec^2sqrtx * d/(dx)[sqrtx])`:

`=> 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, `u = sqrtx`, so now `color(green)((du)/(dx) = 1/(2sqrtx))`:

`=> 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. :-)