Question

Explain the connection between the chain rule for differentiation and the method of u-substitution for integration.

Answer 1

Chain rule:
`(d)/(dx) [f[u(x)]] = (df)/(du) \cdot (du)/(dx)`, u-Substitution:
`f\left[u(x)\right] = \int {(df )/(du) } \, du`

Explanation:

Differentiation and integration are reciprocal to each other. The chain rule indicate that a composite function must be differentiated, describing an inductive approach, whereas u-substitution allows integration by simplifying the expression in a deductive manner. That is:

`(d)/(dx) [f[u(x)]] = (df)/(du) \cdot (du)/(dx)`

Let integrate both sides in terms of x:

`f[u(x)] = \int {(df)/(du) (du)/(dx) } \, dx`

`f\left[u(x)\right] = \int {(df )/(du) } \, du`

This result indicates that f must be rewritten in terms of u and after that first derivative needs to be found before integration.