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

Question

Ezekiel Alexander · Accepted 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, #d/(dx)[f(u(x))] = (df(u))/(du(x))*(du(x))/(dx)#. So:

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. :-)

Paisley Johnson · Answer

undefined To find the derivative of $ e^{2 \tan(\sqrt{x})} $, you would use the chain rule.

The derivative of $ e^{u} $ with respect to $ x $ is $ e^{u} \cdot u' $, where $ u' $ is the derivative of $ u $ with respect to $ x $.

So, in this case, let $ u = 2 \tan(\sqrt{x}) $. Then, $ u' $ (the derivative of $ u $) is $ 2 \sec^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} $.

Now, using the chain rule, the derivative of $ e^{2 \tan(\sqrt{x})} $ with respect to $ x $ is $ e^{2 \tan(\sqrt{x})} \cdot 2 \sec^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} $.

Simplifying, we get $ e^{2 \tan(\sqrt{x})} \cdot \sec^2(\sqrt{x}) \cdot \frac{1}{\sqrt{x}} $.