Chain Rule by decomposition
The Chain Rule applies to any composite function, but we will only deal with composite functions in which the last operation is one of the following:
a trigonometric function (sine, cosine, tangent, etc.), exponent, logarithm, natural logarithm and raise to a power
Any composite function can be written in the form:
y = f(g(x)) = fg(x)
This can be decomposed into y = f(u) where u = g(x).
The Chain Rule says that the derivative of y with respect to the variable x is given by:
The steps are:
This example may help you to follow the chain rule method.
We apply the method to some more examples.
Now lets start with the function:
Differentiate each of these composite functions with respect to x.
A function can be the composition of several functions in succession. We then have to apply the chain rule several times.
Differentiate each equation.