Identify the "inside function" u = f(x) and the "outside function" y = g(u). Then find dy/dx using the Chain Rule. y = sec √(x)

Question

Identify the "inside function" u = f(x) and the "outside function" y = g(u). Then find dy/dx using the Chain Rule.

y = sec
`√(x)`

Answer 1

dfLet
`f(x)=\sec x` and
`g(x)=\sqrt x`. Then


`y=\sec\sqrt x=\sec(g(x))=f(g(x))=f\circ g(x)`

By the chain rule,


`(\mathrm dy)/(\mathrm dx)=(\mathrm dy)/(\mathrm du)(\mathrm du)/(\mathrm dx)`

where
`u=g(x)=\sqrt x`, so that
`y=f(g(x))=f(u)=\sec u`. We have


`(\mathrm du)/(\mathrm dx)=\frac1{2\sqrt x}`

`(\mathrm d\sec u)/(\mathrm du)=\sec u\tan u`

and so


`(\mathrm dy)/(\mathrm dx)=\sec u\tan u(\mathrm dy)/(\mathrm du)=(\sec\sqrt x\,\tan\sqrt x)/(2\sqrt x)`