Question

Use your knowledge of the derivative to compute the value of the limit given below: \lim_{h \rightarrow 0} \frac{\sqrt[7]{(x + h)^{8}} - \sqrt[7]{x^{8}}}{h} .

Answer 1

I just hope I typed this correctly. I copied and pasted from a problem that was similar to mine, I just changed out the numbers I needed. It's for the seventh root of x^8

Answer 2

\[
\lim_{h \rightarrow 0} \frac{\sqrt[7]{(x + h)^{8}} - \sqrt[7]{x^{8}}}{h}
\]

Answer 3

Yea, that's it.

Answer 4

Remember that
\[
f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}
\]And in this case:
\[
f(x) = \sqrt[7]{x^{8}}
\]

Answer 5

I understand that, but I'm unsure how to solve it. If I remember correctly I have to multiply by it's conjugate (I hope that's the correct word), but I keep doing it incorrectly.

Answer 6

You just have to take the derivative....\[ \Large
f(x) = \sqrt[7]{x^8} = x ^{\frac{8}{7}}
\]Where is \(f'(x)\)?

Answer 7

got it, thank you