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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.

The 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 e^u

At times the expression does not appear to be of these, but often it can be written in these forms.
For example, is x⁻²,is e^−2x , is x^½, 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.

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

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:

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 >>