Differentiability of Composite Functions

Let f,g,h be continuous real functions such that: where h′ denotes the derivative of h. Using the Dx notation: This is often informally referred to as the chain rule (for differentiation). Leibniz's notation for derivatives (dydx) allows for a particularly elegant statement of this rule: dydx=dydu⋅dudx where:

• dydx is the derivative of y with respect to x

• dydu is the derivative of y with respect to u

• dudx is the derivative of u with respect to x

However, this must not be interpreted to mean that derivatives can be treated as fractions. It simply is a convenient notation.

dy/dx=(dydu)(dxdu) for dxdu≠0.

2

Thus: There are two cases to consider:

Suppose g′(x)≠0 and that δx is small but non-zero. Then δy≠0 from (1) above, and: hence the result.

Now suppose g′(x)=0 and that δx is small but non-zero. Again, there are two possibilities:

If δy=0, then h(x+δx)−h(x)δx=0. Hence the result.

If δy≠0, then h(x+δx)−h(x)δx=f(y+δy)−f(y)δyδyδx. As δy→0: (1):f(y+δy)−f(y)δy→f′(y) (2):δyδx→0 Thus: limδx→0h(x+δx)−h(x)δx→0=f′(y)g′(x) Again, hence the result. or fog(x) is called a composite function or a function of a function. The theorem for finding the derivative of a composite function is known as the CHAIN RULE.

If f and g are differentiable and are defined by y = f (u) and u = g (x), then the composite function y = f [ g (x) ] is differentiable and we have Corollary : If y = f (u), u = g (v) and v = h (x) where f, g and h are differentiable functions of u, v and x respectively, then

Composite functions are much more common than you may realize. For instance, if you want to compute on your hand-held calculator, you will enter 1.1 and then press the button which squares the entry. After that, you will press the button which exponentiates the entry. Each of the buttons you press, to square and to exponentiate, represent a function. By feeding what comes out of the squaring function into the exponentiation function, you are really computing the composite of these two functions. Said in mathematical notation, you are first evaluating at 1.1 and then putting the result into the function . By composing them, you obtain which you evaluate at 1.1. If we have two functions f(x) and g(x) we can define a composite function h(x)f(g(x)) Thus if f(x)=x3 and g(x)=2x−1 we have h(x)=(2x−1)3=8x3−12x2+6x−1 On the other hand if we define the composite function k(x)g(f(x)) we then have k(x)=2(x3)−1 Notice that h(x) and k(x) are different functions: when composing functions, the order matters. (If you think of functions as procedures, this makes intuitive sense: putting on your socks and then your shoes has a different result from doing this the other way round\dots). In compounding functions such as h(x)=f(g(x)) you have to be a bit careful to ensure that the range of g

Composite functions are what you get when you take the output of one function and use it for the input of the next one. In this discussion, we will discuss the composition of functions which are R1 R1, i.e. a real number as an input and a real number as an output. The notation for this is (fg)(x)=f(g(x)), where the output of g(x) becomes the input of f(x) and is described as (fg)(x). As a real example, let's use f(x)=x2+2*x-2, and g(x)=3*x+2. By replacing all of the occurrences of x in f(x) by the formula for g(x) we can find the formula in x for the composite function. So: (fg)(x) = f(g(x)) = f(3*x+2) (fg)(x) = (3*x+2)2+2*(3*x+2)-2 (fg)(x) = (9*x2+12*x+4)+(6*x+4)-2 (fg)(x) = 9*x2+18*x+6 We can also find (gf)(x): (gf)(x) = g(f(x)) = g(x2+2*x-2) (gf)(x) = 3*(x2+2*x-2)+2 (gf)(x) = 3*x2+6*x-4 That's basic composite functions. Compositing functions can do weird things to the domain and range, though, because your doing mappings onto mappings (i.e. mapping one domain to another domain to a range). One important point is that the composite of a function with its inverse yields an identity function. That is f(f-1(x))=f-1(f(x))=x. What's happening is that your mapping the original function to its range and then straight back into the domain.

• Education Commission of the States. (1994). The Future of Education Policy: An Asian-Pacific/ North America Dialogue 6-7 July 1994. Honolulu , Denver, Colorado

• Edward, T.R. (2007). How to Build A High Performance Organization: A Global Study of Current Trends and Future Possibilities 2007-2017. American Management Association, USA. Pp. 1-6.

• Efthimiou, H. (1999) The Problematic of Comparative Teacher Education: Perspectives from Greece and the United States. Dissertation Abstracts International, 59, (10), April, p.3790-A.Emerald database, http://www.emeraldinsight.com

• The Asian Productivity Organization. Hirakawacho, Chiyoda-ku, Tokyo,Japan Erickson, .R (2001) Morning Conversation: A Study of the Lived Experience of Human Resource Development Practitioner, Dissertation Abstracts International, 62 (5), November, p.1745-A

• Fallon, J.J. (1999) The Impact of In-service Teacher Training on the Writing of 879 High School Juniors in Five west Central Ohio counties. Dissertation Abstracts International, 59 (8), February, p.2139-A.

• Fletcher, C. (1997) Appraisal: Routes to improved performance, 2nd ed. London, IPD.

• Foss, M.J.M (2001) Perceptions of Pos Secondary Precision Machining Instructors;

• Induction, Teacher Preparation, Evaluation and Critical Reflection.

• Dissertation Abstracts International, 61 (11), May, p.4268-A.

• Franks, R.A (2001) An Investigation into the Effectiveness of the “Trainer of Trainers” Model for In-service Science Professional Development

• Programmes for Elementary Teachers. Dissertation Abstracts International Vol. 61 (11), May, p.4342-A.

• Frederick, H. and Charles A. (1964). Myers Education, Manpower and Economic Growth: Strategies of Human Resources Development (McGraw-Hill, USA.

• Friedrich, L.D. (2002) Generating Knowledge for Practice: Joint Work Amount Teacher Professional Development Organizations. Dissertation Abstracts International 62 (10), April, p.3352-A.

 Frost, J.B (2001) A Case Study of the Ways Individuals Conceptualized and Participated in a Professional Development School. Dissertation Abstracts International, 61 (9), April, p.3352-A.