Question

y=$5\sqrt[4]{x}$-$7\sqrt[6]{x}$ найти производную

Answer 1

Решение:

$\displaystyle \Big (y \Big )' = \Big (5 \sqrt[4]{x} - 7 \sqrt[6]{x} \Big )' = \Big (5 x^{\frac {1}{4}} - 7 x^{\frac{1}{6}} \Big )' = 5 \cdot \Big (x^{\frac{1}{4}} \Big )' - 7 \cdot \Big (x^{\frac{1}{6}} \Big )' =\\\\= 5 \cdot \frac{1}{4} \cdot \Big (x^{-\frac{3}{4}} \Big ) - 7 \cdot \frac{1}{6} \cdot \Big (x^{-\frac{5}{6} } \Big ) = \frac{5}{4} \cdot \bigg ( \frac{1}{x^{\frac{3}{4}} } \bigg ) - \frac{7}{6} \cdot \bigg ( \frac{1}{x^{\frac{5}{6}}} \bigg ) =$

$\displaystyle \large {\boxed{ = \;\; \frac{5}{4x^{\frac{3}{4}} } - \frac{7}{6x^{\frac{5}{6}}} \;\; = \;\; \frac{5}{4 \sqrt[4]{x^3} } - \frac{7}{6 \sqrt[6]{x^5} } }}$

Использованные формулы:

• $\Big (x^n \Big )' = n \; x^{n-1}$
• $\Big (k \; f(x) \Big )' = k \; \Big (f(x) \Big )'$
• $\Big ( f(x) + g(x) \Big )' = \Big (f(x) \Big )' + \Big (g(x) \Big )'$

• $\displaystyle x^\frac{a}{b} = \sqrt[b]{x^a}$