Question

Tentukan turunan dari fungsi fungsi berikut
a.f(x)=Cos x(x"-Sinx)
b.f(x)=x"Sec x

Answer 1

(a) u = cos x ---> u' = -sin x
v = x² - sin x ---> v' = 2x - cos x
f(x) = cos x ( x² - sinx ) = u . v
f'(x) = u' v + u v'
= -sin x ( x² - sin x ) + cos x ( 2x - cos x )
= ( -x² sin x + sin² x ) + ( 2x cos x - cos^2 x )
= 2x cos x - x² sin x + ( sin² x - cos² x ) ---> sin² x - cos² x = -cos 2x
= 2x cos x - x² sin x - cos 2x

(b) u = x² ---> u' = 2x
v = cos x ---> v' = -sin x
f(x) = x² sec x = x² / cos x = u / v
f'(x) = ( u' v - u v' ) / v²
= [ 2x . cos x - x² . (-sin x ) ] / cos² x
= [ 2x cos x + x² sin x ) / cos^x
= ( 2x / cos x ) + ( x² tan x / cos x )
= 2x sec x + x² tan x sec x