What is #dy/dx# when #y=(arctanx)^(tanx) , x>0?#
1 Answer
Alternatively:
is equally valid.
Explanation:
For this problem, notice that you have a variable in both your base and exponent. For this reason, you cannot use either the exponent rule (like you would for
Hence, you'll need to do a nifty bit of algebra to solve this. The first step is to take the natural log of both sides, as follows:
Now, you can use a property of logarithms (I have a video on this, if you need a refresher) to bring down the exponent to the front, as follows:
Now, this is simply a chain rule - product rule problem. The only difference is that you have a
All said, after you've taken your derivative (I'm going to assume you know how to do the product rule for the right hand side), you should end up with:
Note that there's a
There's just one final step: solve for
This is a perfectly acceptable final answer. Now, if you'd like to take this one step further and have your final answer in terms of
For this particular problem, plugging in for
That's all! If you'd like some further assistance with this concept, I have a couple of videos you can check out:
[Logarithmic Differentiation Theory Video ()
[Logarithmic Differentiation Practice Problem Video ()
Hope that helps :)