Turunkan masing - masing ekspresi berikut terhadap t. c. t3t−4

Pertanyaan

Turunkan masing - masing ekspresi berikut terhadap t.

c. $\style{font-size:14px}{\frac{3\sqrt t-4}{\sqrt t}}$

Y. Dwi

Master Teacher

Jawaban terverifikasi

Pembahasan

Diketahui $\style{font-size:14px}{\frac{3\sqrt t-4}{\sqrt t}}$ . Ingat untuk $f\left(x\right)=\frac uv$ berlaku $table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction end cell end table$ .

Dimisalkan sebagai berikut.

$u=3\sqrt t-4\\u=3t^\frac12-4\\\\v=\sqrt t\\v=t^\frac12$

Ingat sifat turunan fungsi $ax^n=n\cdot ax^{n-1}$ , maka diperoleh sebagai berikut.

$u'=\frac123t^{\frac12-1}\\u'=\frac32t^{-\frac12}\\\\v'=\frac12t^{\frac12-1}\\v'=\frac12t^{-\frac12}$

Dengan demikian, nilai $f'\left(t\right)$ dapat ditentukan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis t right parenthesis end cell equals cell fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction end cell row blank equals cell fraction numerator open parentheses begin display style 3 over 2 end style t to the power of negative begin display style 1 half end style end exponent times t to the power of begin display style 1 half end style end exponent close parentheses minus open parentheses open parentheses 3 t to the power of 1 half end exponent minus 4 close parentheses times 1 half t to the power of negative 1 half end exponent close parentheses over denominator open parentheses square root of t close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses begin display style 3 over 2 end style t to the power of 0 close parentheses minus open parentheses begin display style 3 over 2 end style t to the power of 0 minus 2 t to the power of negative 1 half end exponent close parentheses over denominator t end fraction end cell row blank equals cell fraction numerator begin display style 3 over 2 end style minus begin display style 3 over 2 end style plus 2 t to the power of negative 1 half end exponent over denominator t end fraction end cell row blank equals cell fraction numerator 2 t to the power of negative 1 half end exponent over denominator t end fraction end cell row blank equals cell 2 t to the power of negative 1 half end exponent times t to the power of negative 1 end exponent end cell row blank equals cell 2 t to the power of negative 3 over 2 end exponent end cell row blank equals cell 2 over t to the power of begin display style 3 over 2 end style end exponent end cell row blank equals cell fraction numerator 2 over denominator t square root of t end fraction end cell row blank blank blank end table$

Dengan demikian, turunannya adalah $fraction numerator 2 over denominator t square root of t end fraction$ .