( 1) y=(tanx)^x

lny=xln(tanx)

1/y * y ' = x ' ln(tanx)+x[ln(tanx)]'

=ln(tanx)+x/tanx*(tanx)'

=lntanx+x/tanx*sec^2x

y'=y(lntanx+x/tanx*sec^2x)

=(tanx)^xlntanx+(tanx)^(x- 1)xsec^2x

X = arctangent

y=ln( 1+t^2)

dx/dt= 1/( 1+t^2)

dy/dt= 1/( 1+t^2)*( 1+t^2)'

= 1/( 1+t^2)*2t

=2t/( 1+t^2)

dy/dx=(dy/dt)/(dx/dt)

=[2t/( 1+t^2)]/[ 1/( 1+t^2)]

=2t

3、

The extreme value of (1) f(x)=x+√( 1-x)

f '(x)= 1+ 1/[2 √( 1-x)]*( 1-x)'

= 1+ 1/[2 √( 1-x)](- 1)

= 1- 1/[2 √( 1-x)]

f'(x)=0

1- 1/[2 √( 1-x)]= 0

2√( 1-x)= 1

√( 1-x)= 1/2

1-x= 1/4

x=3/4

f’(x)>0

1- 1/[2 √( 1-x)]& gt； 0

2√( 1-x)> 1

1-x & gt； 1/4

x & lt3/4

f '(x)& lt； 0，x & gt3/4

When x = 3/4, the maximum value is obtained: f(3/4)=3/4+√( 1-3/4)=5/4.