U-substitution

To differentiate a function of a function, we can use u-substitution. For example,

`(dF(G(x)))/(dx)=??`

We use the substitution,

`u=G(x)`

and differentiate it,

`(du)/(dx)=(dG(x))/(dx)`

Then using the definition of `u`, we differentiate `F(G(x))`,

`F(G(x))=F(u)`

and so,

`(dF(G(x)))/(dx)=(dF(u))/(du) (du)/(dx) = (dF(G(x)))/(dG(x)) (dG(x))/(dx)`

This is a statement of the chain rule.

Find the derivative of `exp(lambdax)`.

Using the chain rule we have,

`G(x)=lambdax` and `F(G(lambdax))=F(G(u))`

Thus,

`(dexp(lambdax))/(dx)=(dexp(u))/(du) (du)/(dx)`

Let `u=lambdax`, then `(du)/(dx)=lambda` and so,

`(dexp(u))/(du) (du)/(dx) = lambdaexp(u)`

We then reverse our substitution for `u`, and obtain,

`(dexp(lambdax))/(dx)=lambdaexp(lambdax)`