How to Differentiate a Function to the Power of a Function

Sudhanshu asked me to produce a video explaining how to differentiate a function to the power of another function, in other words, y = f(x)^g(x). This is the result!

Rather than provide all the theory, I demonstrate the process using three functions:
y = cosθ^tanθ
y = ln(x²)^sin(2x)
y = x^x

In each case we convert the function f(x) into an exponential function by using the fact that exponentials and logarithms are inverse functions. This means that f(x) = e^ln[f(x)]

Once this conversion has been made, we now have to find the derivative of an exponential function y = e^{ln[f(x)].g(x)} ... and "its bark is worse than its bite!" In other words, it is easier than it looks!

This requires the chain rule, since we have a function (exponential base e) of a function {ln[f(x)].g(x)}. Note that we will also require the product rule because ln[f(x)].g(x) is a product of two functions.

Thank you for asking this question, Sudhanshu. I hope the video answers your question and helps others as well.

If you want a good variety of examples to practise your skills in finding derivatives, including some of the form f(x)^g(x), download my FREE PDF FILE which should give you sufficient revision practice for quite a while. You might like to work through them either on your own or with friends. You may download the file from here: http://crystalclearmaths.com/wp-conte....

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Best wishes with your study and your mathematics!

Thank you.