rationalskepticism.org

This is the last differentiation rule you will need to know. What happens when we have nested functions? That means a function like:



or with an example:



Well, the rule says that the derivative will be equal to the derivative of the f(x) times the derivative of the g(f(x)). Let's see it in mathematical terms and with our example in action:



In our example we have:




So their derivatives will be:




Then combining the two, the derivative will be:



Despite this one is seemingly difficult, it is very very easy to remember and to differentiate these kinds of functions.



If you have a function in more than one variables, let's say f(x,t), and your problem says to differentiate with respect to t (for example), then you treat the x variable as a constant. The same applies for functions of more variables; you differentiate the variable you are told, and you treat the others as constants.

When you have a function made of multiple functions, remember to identify each function separately, and first differentiate those. Then apply one by one the appropriate rules, and start combining the results. Especially at the beginning do not try to do it all at once.

Remember to identify which is the variable and which are constants; sometimes you have a general problem like [math]f(x)=x^n, and you are told that n is a constant. It may be that you may have more than one letters acting as constants.

In this short tutorial, I have mostly used x as the independent variable and y as the dependent variable; in the real world, this may change and you may have different combinations of letters. For example, in physics it is common to t for time, and to differentiate with respect to t. There are a number of other variables used.

I will try to start with a few examples tomorrow, more complex than the ones I have put here.