OBLICZANIE PROSTYCH POCHODNYCH - ZADANIA
Zadanie 5
Oblicz pochodne funkcji: \(\displaystyle f(x)=\frac{1}{\sqrt x}\), \(\displaystyle g(x)=\frac{1}{\sqrt {x^3}}\), \(\displaystyle h(x)=\frac{1}{\sqrt[3]x}\), \(\displaystyle k(x)=\frac{1}{\sqrt[4]{x^7}}\)
Do obliczania pochodnych powyższych funkcji będziemy stosować wzór:
\(\displaystyle (x^n)'=nx^{n-1} \)
\(\displaystyle f'(x)=\left(\frac{1}{\sqrt x}\right)'=\left(\frac{1}{x^{\frac12}}\right)'= \left({x^{-\frac12}}\right)'=\) \(\displaystyle-\frac12{x^{\left(-\frac12-1\right)}}= {-\frac12x^{\left(-\frac32\right)}}=-\frac12\frac{1}{x^{\frac32}}=\) \(\displaystyle \frac{-1}{2\sqrt {x^3}} \)
\(\displaystyle g'(x)=\left(\frac{1}{\sqrt {x^3}}\right)'=\left(\frac{1}{x^{\frac32}}\right)'= \left({x^{-\frac32}}\right)'=\) \(\displaystyle-\frac32{x^{\left(-\frac32-1\right)}}= {-\frac32x^{\left(-\frac52\right)}}=-\frac32\frac{1}{x^{\frac52}}=\) \(\displaystyle \frac{-3}{2\sqrt {x^5}}\)
\(\displaystyle h'(x)=\left(\frac{1}{\sqrt[3]x}\right)'=\left(\frac{1}{x^{\frac13}}\right)'= \left({x^{-\frac13}}\right)'=\) \(\displaystyle-\frac13{x^{\left(-\frac13-1\right)}}= {-\frac13x^{\left(-\frac43\right)}}=-\frac13\frac{1}{x^{\frac43}}=\) \(\displaystyle \frac{-1}{3\sqrt[3]{x^4}}\)
\(\displaystyle k'(x)=\left(\frac{1}{\sqrt[4]{x^7}}\right)'=\left(\frac{1}{x^{\frac47}}\right)'= \left({x^{-\frac47}}\right)'=\) \(\displaystyle-\frac47{x^{\left(-\frac47-1\right)}}= {-\frac47x^{\left(-\frac{11}7\right)}}=-\frac47\frac{1}{x^{\frac{11}4}}=\) \(\displaystyle \frac{-4}{7\sqrt[4]{x^{11}}}\)
