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Chain Rule

Inner and Outer Functions

In general, by substituting the function g(x) for u in the function f(u) we get the composite function h(x) = f(g(x)).

Examples

Let f(u) = sin u. If we substitute x2 for u, we get the composite function h(x) = f(x2) = sin (x2)

Let f(u) = u10. If we substitute x2 + 1 for u we get the composite function h(x) = f(x2 + 1) = (x2 + 1)10

Let f(u) = u5. If we substitute sin (x2 + 1) for u we get the composite function h(x) = f(sin (x2 + 1)) = (sin (x2 + 1))5 which is often written sin5 (x2 + 1)

We call g the inner function, and f the outer function of the composition. g may be any function, and often is itself another composite function.

sock and bootThe reversed process of composition is called decomposition. Composition is like dressing your feet, socks on first then boots on, and decomposition like undressing, you take boots off first then socks off.

Some composite functions can be decomposed in several ways. We will only consider those functions whose outermost function is a basic function of the form

un, sin u, cos u, tan u, ln u and eu

At times the expression does not appear to be of these, but often it can be written in these forms.
For example,1 over x squared is x−2,1 over e to the power 2xis e−2x ,1 over e to the power 2x is x½, sin2 x is (sin x)2, cos3 (x2 + 1) is (cos (x2 + 1))3

Look now at a decomposition. We need only decompose until we reach a simple function, which we can differentiate easily.

worked

The order of decomposition is very important. Notice the difference in decomposition in the next two examples.

y equals the cosine of (x squared) decomposes to y equals the cosine of u and u equals x squared

 

y equals the cosine of x all squared decomposes to y equals u squared and u equals the cosine of x

Examples

h(x) = (x2 + 1)10 has outer function y = f(u) = u10 and inner function u = g(x) = x2 + 1.

h(x) = log (cos x) has outer function y = f(u) = log u and inner function u = g(x) = cos x.

Exercise

Complete the table:

Function
h(x)

Outer function
f(u)

Inner function
g(x)

1.

cos (x2 + 1)

outer function is cos u

inner function is (x^2)+1

Check

2.

ln (sec x)

outer function is ln(u

inner function is sec(x)

Check

3.

ex

outer function is exp(u)

inner function is x*(−1)

Check

4.

tan3 (5/x)

outer function is u^3

inner function is tan(5/x)

Check

5.

(log (2x))−2

outer function is u^(-2)

inner function is log(2x)

Check

If you are finding these problems difficult, a review of algebraic order of operations may help.

<< Composite Functions | Chain Rule Index | Chain Rule by decomposition >>

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