Question

what is \\(\lim_{h \to 0} \frac{8\left(\frac{1}{2} + h\
ight)^8 - 8\left(\frac{1}{2}\
ight)^8}{h}\\)?

Explanation:

Step1: Recognize derivative definition

Recall that $f'(a)=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$. Here, $f(x)=8x^{\frac{1}{8}}$, $a=1$.

Step2: Compute derivative of $f(x)$

Use power rule: $\frac{d}{dx}[x^n]=nx^{n-1}$.
$f'(x)=8\times\frac{1}{8}x^{\frac{1}{8}-1}=x^{-\frac{7}{8}}$

Step3: Evaluate at $x=1$

Substitute $x=1$ into $f'(x)$:
$f'(1)=1^{-\frac{7}{8}}=1$

Answer:

1