With some functions and expressions, differentiation is a long process. e.g. Differentiating the function h(x) = (2x + 3)5 first needs a long expansion of the brackets.
If we think of h(x) = (2x + 3)5 as a combination of two functions with g(x) = 2x + 3 being inside the function f(x) so that
f(x) = (g(x))5. This is known as a function of a function or a composite function f o g.
To differentiate a function of a function we use the Chain Rule.
Chain Rule Formula
(f o g )'(x) = f '(g(x)) . g'(x)
The Chain Rule looks complex, but with practice, which is essential, it should become more straightforward.
Example For our example, to differentiate h(x) = (2x + 3)5:
Let f(x) = x5 and g(x) = 2x + 3
Therefore, f '(x) = 5x4 and g '(x) = 2
h '(x) = (f o g )'(x) = f '(g(x)) . g'(x)
= 5(2x + 3)4 . 2
= 10(2x + 3)4
Alternative Chain Rule formula
Another way to differentiate a composite function is to use notation, which behaves like fractions.
The Chain Rule states:
Example For our example, to differentiate y = (2x + 3)5
Let u = 2x + 3, so y = u5 Therefore, = 2 and = 5u4 Using the chain rule = 5u4 . 2 substituting u back in. = 10(2x + 3)4
Roots and Negative Indices using the Chain Rule
As with most differentiation, rational functions and expressions involving roots must be written in index form before differentiation.
Write the function as y = (x2 + 3x)-1
Differentiate f(x) = √(3x2 + 2)
Write the expression as f(x) =