Question

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]

Answer 1

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]

Answer 2

PEMBAHASAN

Trigonometri

y = sec ax → y' = a sec ax tan ax

y = tan ax → y' = a sec² ax

π/12 = 180°/12 = 15°

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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