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Jika y = tanh u dan u = f(x) , maka dy du<br />
= sech u<br />
dx dx<br />
(5.50)<br />
Bukti<br />
y = tanh u ® dy<br />
du = sec u<br />
dy<br />
dx = dy du du<br />
= sech u<br />
du dx dx (terbukti)<br />
Contoh 5.27<br />
Jika y = tanh(a + bx), tentukan dy<br />
dx<br />
Penyelesaian<br />
Misal u = a+bx y = tanh u<br />
du<br />
dx = b<br />
dy<br />
du = sech u<br />
dy<br />
dx = dy du<br />
= (sech u)(b) = b sech (a + bx)<br />
du dx<br />
Jika y = f(x) = coth x , maka dy = f'(x) = −csch x (5.51)<br />
dx<br />
Bukti<br />
y = f(x) = cothx = coshx<br />
sinhx<br />
dy<br />
(sinhx)(sinhx) − (coshx)(coshx)<br />
= f'(x) = =<br />
dx (sinhx)<br />
= −1 = −csch x (terbukti)<br />
sinh x<br />
sinh x − cosh x<br />
sinh x<br />
Jika y = coth u dan u = f(x) , maka dy<br />
du<br />
= −csch u<br />
dx dx<br />
(5.52)<br />
Bukti<br />
y = tanh u ® dy<br />
du = −csch u<br />
dy<br />
dx = dy du<br />
du<br />
= −csch u<br />
du dx dx (terbukti)<br />
Contoh 5.28<br />
Jika y = coth(a + bt), tentukan dy<br />
dt<br />
Penyelesaian<br />
Misal u = a+bt<br />
y = coth u<br />
du<br />
dx = b<br />
dy<br />
du = −csch u<br />
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