📋 Daftar Isi
Aturan-aturan
Beberapa aturan yang digunakan pada turunan fungsi trigonometri, sebagai berikut :
.tg-wrap {padding-botom: 20px;} .tg {text-align:center;vertical-align:top;border-collapse:collapse;border-color:#ccc;border-spacing:0;} .tg td{border-bottom-width:1px;border-color:black;border-style:solid;border-top-width:1px;border-width:0px; font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-bottom-width:1px;border-color:black;border-style:solid;border-top-width:1px;border-width:0px; font-family:Arial, sans-serif;font-size:14px;font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-0lax{text-align:left;vertical-align:middle} .tg .tg-awal{width:50%;text-align:left;vertical-align:middle} .tg .tg-tf2e{text-align:left;vertical-align:middle} .tg-tf2e .mjx-chtml {text-align: center !important;}| f(x) | f'(x) |
|---|---|
| \[\sin U\] | \[\cos U \cdot U’\] |
| \[\cos U\] | \[-\sin U \cdot U’\] |
| \[\tan U\] | \[\sec^{2}U \cdot U’\] |
| \[\sec U\] | \[\sec U \cdot \tan U \cdot U’\] |
| \[\cot U\] | \[-\csc^{2}U \cdot U’\] |
| \[\csc U\] | \[\csc U·cot U \cdot U’\] |
Contoh Soal dan Penyelesaian
Pengerjaan bentuk U + V dan U – V
Contoh 1
Tentukan turunan dari
\[
f(x)=2 \cos x – \sin 4x + \tan x
\]
Jawab 1
\[
f'(x) = -2 \sin x – 4 \cos 4x + \sec^{2}x
\]
Contoh 2
Tentukan turunan dari
\[
h(x) = \cos x + x \sin x – x^{3}+5
\]
Jawab 2
\[
h'(x)= – \sin x + (1) (\sin x) + (1)(\cos x) – 3x^{2}+0
\]
\[
h'(x)= – \sin x + \sin x + x \cos x – 3x^{2}
\]
\[
h'(x)= x \cos x – 3x^{2}
\]
Pengerjaan bentuk U · V
Contoh 1
Tentukan turunan pertama dan kedua dari
\[
y=(\sin x – \cos x)(\sin x + \cos x)
\]
Jawab 1
\[
U= \sin x – \cos x
\]
\[
V= \sin x + \cos x
\]
\[
U’= \cos x + \sin x
\]
\[
V’= \cos x – \sin x
\]
\[
y’= U’V + UV’
\]
\[
y’=(\cos x + \sin x)(\sin x + \cos x)+(\sin x – \cos x)(\cos x – \sin x)
\]
\[
y’= \sin^{2} x + 2 \sin x \cos x + \cos^{2} x – (\sin^{2} x – 2 \sin x \cos x + \cos^{2}x)
\]
\[
y’=4 \sin x \cos x
\]
\[
y’=2 \sin 2x
\]
\[
y”=4 \cos 2x
\]
Contoh 2
Tentukan turunan dari
\[
y=4 \sin^{2} x \cos 2x
\]
Jawab 2
\[
U= 4 \sin^{2}x
\]
\[
V= \cos 2x
\]
\[
U’=4(2 \sin x \cos x)
\]
\[
U’=4 \sin 2x
\]
\[
V’=-2 \sin 2x
\]
\[
y’=U’V+UV’
\]
\[
y’=(4 \sin 2x)(\cos 2x)+(4 \sin^{2} x)(-2 \sin 2x)
\]
\[
y’=2 \sin 4x – 8 \sin^{2} x \sin 2x
\]
Pengerjaan Bentuk U/V
Contoh 1
Tentukan nilai y’ dari
\[
y=\frac{\sin x}{1- \cos x}
\]
Jawab 1
\[
U=\sin x
\]
\[
V=1-\cos x
\]
\[
U’=\cos x
\]
\[
V’=\sin x
\]
\[
y’=\frac{U’V-UV’}{V^{2}}
\]
\[
y’=\frac{(\cos x)(1-\cos x)-(\sin x)(\sin x)}{(1-\cos x)^{2}}
\]
\[
y’=\frac{\cos x-\cos^{2} x-\sin^{2} x}{(1-\cos x)(1-\cos x)}
\]
\[
y’=\frac{-(-\cos x+(\cos^{2} x+\sin^{2} x))}{(1-\cos x)(1-\cos x)}
\]
\[
y’=\frac{-(-\cos x+x)}{(1-\cos x)(1-\cos x)}
\]
\[
y’=\frac{1}{\cos x-1}
\]
Contoh 2
Tentukan turunan dari
\[
f(x)=\frac{x+sinx}{1+cosx}
\]
Jawab 2
\[
U=x+sinx
\]
\[
V=1+cosx
\]
\[
U’=1+cosx
\]
\[
V’=-sinx
\]
\[
f'(x)=\frac{U’V-UV’}{V^{2}}
\]
\[
f'(x)=\frac{(1+cosx)(1+cosx)-(xsinx)(-sinx)}{(1+cosx)^{2}}
\]
\[
f'(x)=\frac{1+2cosx+cos^{2}x+xsinx+sin^{2}x}{(1+cosx)^{2}}
\]
\[
f'(x)=\frac{2+xsinx+2cosx}{(1+cosx)^{2}}
\]
Pengerjaan Bentuk Un
Contoh 1
Tentukan turunan dari
\[
y=\sin^{7}(5x^{2}-\frac{\pi}{2})
\]
Jawab 1
\[
y’=n(U^{n-1})U’
\]
\[
y’=7 \sin^{7-1}(5x^{2}-\frac{\pi}{2}) \cos(5x^{2}-\frac{\pi}{2})(2(5x^{2-1}-0)
\]
\[
y’=70x \sin^{6}(5x^{2}-\frac{\pi}{2}) \cos(5x^{2}-\frac{\pi}{2})
\]
Contoh 2
Tentukan turunan dari
\[
f(x)=\sec^{10}(3-5x)
\]
Jawab 2
\[
f'(x)=10 \sec^{10}(3-5x) \sec(3-5x) \tan(3-5x)(-5)
\]
\[
f'(x)=-50 \sec10(3-5x) \tan(3-5x)
\]
