Jawaban:

1) f'(x)=(3/2)√x

2) f'(x)=-1/(x+1)²

Penjelasan dengan langkah-langkah:

definisi turunan dengan menggunakan limit

menggunakan limit

1) f(x)=x√x

f(x)=x.x^(1/2)

f(x)=x^(3/2)

f'(x)

=lim h→0 (f(x+h)-f(x))/h

=lim h→0 ((x+h)^(3/2) -x^(3/2))/h

=lim h→0 ((√(x+h)³ -√(x)³)/h).((√(x+h)³ +√(x)³)/(√(x+h)³ +√(x)³))

=lim h→0 ((x+h)³-(x)³)/(h(√(x+h)³ +√(x)³))

=lim h→0 (x³+3hx²+3xh²+h³-x³)/(h(√(x+h)³ + √(x)³))

=lim h→0 (3hx²+3xh²+h³)/(h(√(x+h)³ +√(x)³))

=lim h→0 (h(3x²+3xh+h²))/(h(√(x+h)³ +√(x)³))

=lim h→0 (3x²+3xh+h²)/(√(x+h)³ +√(x)³))

=lim h→0 (3x²+3x(0)+(0)²)/(√(x+0)³ +√(x)³))

=lim h→0 (3x²)/(√x³ +√x³)

=lim h→0 (3x²)/(2√x³)

=lim h→0 (3/2)x^(2-(3/2))

=(3/2)√x

2) f(x)=(1/(x+1))

f'(x)

=lim h→0 (f(x+h)-f(x))/h

=lim h→0 (1/((x+h)+1) -(1/(x+1)))/h

=lim h→0 (((x+1)-(x+h+1))/((x+h+1)(x+1)))/h

=lim h→0 (-h/(x+h+1)(x+1))/h

=lim h→0 (-1/(x+h+1)(x+1))

=lim h→0 (-1/(x+0+1)(x+1))

=(-1/(x+1)²)

untuk soal perkalian atau pembagian dengan u/v ada aturannya, namanya aturan rantai

f(x)=u.v

f'(x)=u'.v+u.v'

f(x)=u/v

f'(x)=(u'.v-u.v')/v²

saya ambil contoh dengan soal no.1

f(x)=x√x

u=x

u'=1

v=x^(1/2)

v'=1/(2√x)

f'(x)=u'.v+u.v'

f'(x)=1(√x)+x.1/(2√x)

f'(x)=√x + (x/2√x)

f'(x)=(2x+x)/(2√x)

f'(x)=(3x)/(2√x)

f'(x)=(3/2)x^(1-(1/2))

f'(x)=(3/2)√x

semoga dapat dipahami dan bermanfaat