OBLICZANIE PROSTYCH POCHODNYCH - ZADANIA
Zadanie 4
Oblicz pochodne funkcji: \(\displaystyle f(x)=\sqrt x\), \(\displaystyle g(x)=\sqrt {x^3}\), \(\displaystyle h(x)=\sqrt[3]x\), \(\displaystyle k(x)=\sqrt[4]{x^7}\)
Do obliczania pochodnych powyższych funkcji będziemy stosować wzór:
\(\displaystyle (x^n)'=nx^{n-1} \)
\(\displaystyle f'(x)=(\sqrt x)'=(x^{\frac12})'=\frac12 x^{\frac12-1}=\)\(\displaystyle\frac12 x^{-\frac12}=\frac12 \frac1{x^{\frac12}}= \frac1{2\sqrt x}\)
\(\displaystyle g'(x)=(\sqrt {x^3})'=(x^{\frac32})'=\frac32 x^{\frac32-1}=\)\(\displaystyle\frac32 x^{\frac12}=\frac32 \sqrt x\)
\(\displaystyle h'(x)=\left (\sqrt[3]x\right )'=(x^{\frac13})'=\frac13 x^{\frac13-1}=\)\(\displaystyle\frac13 x^{-\frac23}=\frac13 \frac1{x^{\frac23}}= \frac1{3\sqrt[3] {x^2}}\)
\(\displaystyle k'(x)=\left (\sqrt[4]{x^7}\right )'=(x^{\frac74})'=\frac74 x^{\frac74-1}=\)\(\displaystyle\frac74 x^{\frac34}=\frac74 {x^{\frac34}}=\frac74 {\sqrt[3] {x^4}}\)
