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du
dx = 3 4
dy
dx = dy du
du
dx = 1
√u − 1
dy
du = 1
√u − 1
3
4 = 3
4√u − 1 = 3
4
=
3
4 x − 1
3
4 9
16 x − 1
Jika y = f(x) = tanh
x , maka dy
dx = f'(x) = 1
1 − x
, |x| < 1 (5.61)
Bukti
y = f(x) = tanh x = 1 2 ln 1 + x
1 − x , |x| < 1
dy
dx = 1 2 1 − x
2 (1 − x) 1 + x = 1
1 − x , |x| < 1 ( )
Jika y = tanh
Bukti
y = tanh
dy
dx = dy
u ® dy
du = 1
1 − u
dx = 1
1 − u
du
du
Contoh 5.33
u dan u = f(x) , maka dy
dx = 1
1 − u
du
dx , |u| < 1 ( )
Jika y = tanh (2x − 1), tentukan dy
dx
Penyelesaian
Misal u = 2x − 1 y = tanh u
du
dx = 2
dy
du = 1
1 − u
dy
dx = dy du
du dx = 1
2
(2) =
1 − u 1 − (2x − 1)
du
dx
, |u| < 1 (5.62)
Jika y = f(x) = coth
Bukti
x , maka dy
dx = f'(x) = 1 , |x| > 1 (5.63)
x − 1
y = f(x) = coth x = 1 2 ln x + 1
x − 1 , |x| > 1
dy
dx = 1 −2 1 − x
2 (1 − x) 1 + x = − 1
1 − x = 1
x − 1 , |x| > 1 ( )
Jika y = coth
u dan u = f(x) , maka dy
dx = 1
u − 1
du
, |u| > 1 (5.64)
dx
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