a-maths~!~!~

(1)
y = loga (x + x^3)
(2)
y = 2sinx
take ln on both side
ln y = (ln 2) sin x
differentiate w.r.t. x
1/y (dy/dx) = (ln 2) cos x
(dy/dx) = (ln 2) (cos x) y = (ln 2) (cos x) 2sin x
(3)
y = e arcsin √x
dy/dx = (1 / 2√x) (1 / √( 1-x^2)) (e arcsin √x)
= (e arcsin √x / {2√ [x(1 - x^2)]}
(4)
y = ex cos (x/2)
dy/dx = ex (-1/2) sin (x/2) + ex cos (x/2)
= ex [cos (x/2) - 0.5 sin (x/2)]
(5)
y = ln [x + √(1 + x^2)]
dy/dx = {1 + [2x / (2 √(1 + x^2)]}/ [ x + √(1 + x^2)]
= [ x + √(1 + x^2)] / {[√(1 + x^2) ] [x + √( 1+ x^2)]
= 1 / √(1+ x^2)
(6)
y = arctan √x
tan y = √x --> cos y = 1/√(x + 1)
(dy/dx) (sec y)2 = 1 / (2√x)
dy/dx = (cos y)2 / 2√x
dy/dx = 1 / [2(x+1)√x]