Suffice it to say that it is not a trivial exercise. If you are not familiar with the techniques used for factorising polynomials, the page entitled "Polynomials" in the Algebra section might be of interest. factorising the polynomial expression 8 x 3 - 36 x 2 + 54 x - 27) we need to carry out something called a rational root test, because the degree of the polynomial is greater than two ( 2). It is also possible factorise the polynomial expression inside the brackets:įor this last bit of the factorisation (i.e. Anyway, having multiplied out the brackets, and assuming we haven't made any mistakes, we can now apply the basic rules of differentiation to the result to find the derivative: You can probably imagine how easy it is to make an error with this kind of calculation. Bear in mind also that this is a relatively trivial example. We did tidy things up a bit though, rather than show every step in the process. Now we multiply together the resulting trinomials: We start by multiplying together the two pairs of binomials: You might assume that we can simply multiply out the brackets and then apply the basic rules of differentiation in the normal way. Let's suppose that we want to find the derivative of the function ƒ( x) = (2 x - 3) 4. We will see in due course how the chain rule can be applied to a composite function consisting of more than two functions, but for now we will concentrate on composites that involve just two.Ĭomposite functions are nested at different levels, like Russian dollsĪlthough it is possible in theory to find the derivative of composite functions without using the chain rule, this is usually very difficult to achieve in practice. Function g( x) is the inner doll, and function ƒ( x) is the outer doll. It might be helpful here to think of these functions as being like Russian dolls. We can express this relationship formally as follows: First of all we are squaring x, and then we are taking the cosine of the result ( x 2). We are applying the trigonometric function ƒ( x) = cos( x) to the function g( x) = x 2. Supposing we have two functions, ƒ( x) = cos( x) and g( x) = x 2. In other words, it enables us to differentiate an expression called a composite function, in which one function is applied to the output of another. The chain rule gives us a formula that enables us to differentiate a function of a function.
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