Question

Tentukan turunan dari:
1. Y= sinh³(x/3)
2. Y= arg sinh (tg x)
3. Y= In In cosh x
4. X= cosh (In y)

Answer 1

1. subtitusi x/3 = u

y = sinh³(u)

dy = 3.cosh²(u) . du <= u = x/3

   = 3.cosh²(x/3) . d(x/3)

   = 1/3.3.cosh²(x/3) . dx

dy/dx = cosh²(x/3)

2. y = arc sinh(tg x) <= subtitusi tg x = u

y = arc sinh u  <= turunan arc sinh u = 1/√(1+u²)

dy = 1/√(1+u²)  . du

dy = 1/√(1+tg² x) . d(tg x)

dy/dx = sec²x/√(1-tg² x) <= tan²x+1 = sec²x

dy/dx = sec²x/secx

dy/dx = sec x

3. y = ln(ln(cosh(x)) )  <= subtitusi cosh(x) = v , ln(v) = u

y = ln(u) <= turunan ln u = 1/u

dy = 1/u . du <= u = ln(v)

dy = 1/(ln(v)) . d(ln(v))

dy = 1/(ln(v)) . 1/v . dv <= v = cosh(x)

dy = 1/(ln(cosh(x) ) ) . 1/(cosh(x)) . d(cosh(x))

dy/dx = <= sin/cos = tan

dy/dx =

4. x = cosh(ln y) <= subtitusi ln y = u

x = cosh(u) <= turunan cosh(u) = sinh(u) <= beda sedikit karena gak ada tanda minus

dx = sinh(u) . du <= u = ln y

    = sinh(ln y) . d(ln y)

   = 1/y . sinh(ln y).dy

dx/dy =