Tentukan turunan pertama dari [tex]\displaystyle f(x)=\frac{\sec x+\csc x}{\sec x-\csc x}[/tex] lalu hitung [tex]\displaystyle f'\left ( \frac{\pi}{12} \right )[/tex]
" Life is not a problem to be solved but a reality to be experienced! "
© Copyright 2013 - 2024 KUDO.TIPS - All rights reserved.
Verified answer
Penjelasan dengan langkah-langkah:
Fungsi Trigonometri
[tex]\displaystyle f(x)=\frac{\sec x+\csc x}{\sec x-\csc x}[/tex]
.
sec x = [tex]\sf\frac{1}{cos(x)}[/tex]
csc x = [tex]\sf\frac{1}{sin(x)}[/tex]
.
[tex] \sf\frac{1}{cos(x)}\pm\frac{1}{sin(x)}=\frac{sin(x)\pm cos(x)}{sin(x).cos(x)}[/tex]
[tex]\displaystyle \: \sf \: f(x) = \frac{ \frac{ \sin(x) + \cos(x) }{ \sin(x) \cos(x) } }{\frac{ \sin(x) - \cos(x) }{ \sin(x) \cos(x)}} [/tex]
[tex]\displaystyle\sf \: f(x) = \frac{ \sin(x) + \cos(x) }{ \sin(x) - \cos(x) } [/tex]
.
sin (x) + cos (x) = u
sin (x) - cos (x) = v
.
f(x) = u/v
f'(x) = [tex]\sf \frac{u'v-uv'}{v²} [/tex]
.
u = sin (x) + cos (x)
u' = cos (x) - sin (x) => -(sin(x)-cos(x))
v = sin (x) - cos (x)
v' = cos (x) + sin (x)
.
u'v = -(sin(x)-cos(x))×(sin(x)-(cos(x)
u'v = -(sin(x)-cos(x))²
=> (sin (x)±cos (x))² = 1 ± sin (2x)
u'v = -(1 - sin (2x))
u'v = -1 + sin (2x))
.
uv' = (sin (x) + cos (x))×(cos (x) + sin (x))
uv' = (sin(x)+cos(x))²
uv' = 1 + sin (2x)
.
[tex]\displaystyle\sf \: f'(x) = \frac{ - 1 + \sin(2x) - (1 + \sin(2x) }{( \sin(x) - \cos(x) {)}^{2} } [/tex]
[tex]\displaystyle\sf \: f'(x) = \frac{ - 2 }{1 - \sin(2x) } [/tex]
[tex]\displaystyle\sf \: f'(12) = \frac{ - 2 }{1 - \sin(2\left( \frac{\pi}{12} \right) } [/tex]
[tex]\displaystyle\sf \: f'(12) = \frac{ - 2 }{1 - \sin( \frac{\pi}{6} ) } [/tex]
[tex]\displaystyle\sf \: f'(12) = \frac{ - 2 }{1 - \sin( {30}^{\circ} ) } [/tex]
[tex]\displaystyle\sf \: f'(12) = \frac{ - 2 }{1 - \frac{1}{2} } [/tex]
[tex]\displaystyle\sf \: f'(12) = \frac{ - 2 }{ \frac{1}{2} } [/tex]
[tex]\displaystyle\sf \: f'(12) = - 2 \times \frac{2}{1} [/tex]
[tex]\boxed{\displaystyle\sf \: f'(12) = -4}[/tex]
PEMBAHASAN
Trigonometri
y = sec ax → y' = a sec ax tan ax
y = tan ax → y' = a sec² ax
π/12 = 180°/12 = 15°
__
soal
pembilang dan penyebut kalikan sin x cos x
= (sin x - cos x)/(sin x + cos x)
= (sin x - cos x)²/(sin² x - cos² x)
= (1 - sin 2x)/(- cos 2x)
= - sec 2x - tan 2x
f(x) = - sec 2x - tan 2x
f'(x) = - 2 sec 2x tan 2x - 2 sec² 2x
f'(π/12) = f'(15°)
= - 2 sec 30° tan 30° - 2 sec² 30°
= - 2 (2/√3)(1/√3) - 2(2/√3)²
= -4/3 - 8/3
= -12/3
= -4