Question

what is \\( \lim _ { h \
ightarrow 0 } \frac { 8 ( \frac { 1 } { 2 } + h ) ^ { 8 } - 8 ( \frac { 1 } { 2 } ) ^ { 8 } } { h } \\)

Explanation:

Step1: Recognize derivative definition

The limit is $\lim_{h\to0}\frac{f(\frac{1}{2}+h)-f(\frac{1}{2})}{h}$ where $f(x)=8x^8$.

Step2: Compute derivative of $f(x)$

$f'(x)=8\times8x^7=64x^7$.

Step3: Substitute $x=\frac{1}{2}$

$f'(\frac{1}{2})=64\times(\frac{1}{2})^7=64\times\frac{1}{128}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$