if-y-x-1-7-prove-that-y-1-7-x-6-1-7-

Question Number 205013 by mathlove last updated on 05/Mar/24

i f y = \sqrt[7]{x} p r o v e t h a t

y^{^{'}} = \frac{1}{7 \sqrt[7]{x^{6}}}

Answered by Frix last updated on 05/Mar/24

y = x^{r} \Rightarrow y^{'} = {r x}^{r - 1}; r \neq 0

y = \sqrt[7]{x} = x^{\frac{1}{7}} \Rightarrow y^{'} = \frac{1}{7} x^{- \frac{6}{7}} = \frac{1}{7 \sqrt[7]{x^{6}}}

Answered by MM42 last updated on 06/Mar/24

f^{'} (x) = {l i m}_{h \to 0} \frac{\sqrt[7]{x + h} - \sqrt[7]{x}}{h}

= {l i m}_{h \to 0} \frac{h}{h (\sqrt[7]{{(x + h)}^{6}} + \sqrt[7]{{(x + h)}^{5} x} + \dots + \sqrt[7]{x^{6}})}

= \frac{1}{7 \sqrt[7]{x^{6}}} ✓

Commented by mathlove last updated on 06/Mar/24

t h a n k s

Commented by TonyCWX08 last updated on 06/Mar/24

W r o n g!

D e r i v a t i v e o f \sqrt[7]{x}

= \frac{1}{7 \sqrt[7]{x^{6}}}