$\text { Find the derivative of the function } f ( x ) = \frac { 6 } { \sqrt [ 2 ] { x } } \text { by the limit process. }$
A) $f ^ { t } ( x ) = - \frac { 6 } { x ^ { 2 } \sqrt { x } }$
B) $f ^ { t } ( x ) = \frac { 3 } { x }$
C) $f ^ { t } ( x ) = - \frac { 3 \sqrt [ 2 ] { x } } { x }$
D) $f ^ { t } ( x ) = - \frac { 3 } { x ^ { 2 } \sqrt { x } }$
E) $f ^ { t } ( x ) = \frac { 6 } { x }$

Question

$\text { Find the derivative of the function } f ( x ) = \frac { 6 } { \sqrt [ 2 ] { x } } \text { by the limit process. }$ 
A) $f ^ { t } ( x ) = - \frac { 6 } { x ^ { 2 } \sqrt { x } }$
B) $f ^ { t } ( x ) = \frac { 3 } { x }$
C) $f ^ { t } ( x ) = - \frac { 3 \sqrt [ 2 ] { x } } { x }$
D) $f ^ { t } ( x ) = - \frac { 3 } { x ^ { 2 } \sqrt { x } }$
E) $f ^ { t } ( x ) = \frac { 6 } { x }$

Quizplus · Accepted Answer

Using the limit process, we have:

$$f'(x) = \lim_{h	o 0} \frac{f(x+h)-f(x)}{h}$$ 
$$= \lim_{h	o 0} \frac{\frac{6}{\sqrt{x+h}}-\frac{6}{\sqrt{x}}}{h}$$ 
$$= \lim_{h	o 0} \frac{6\sqrt{x}-6\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}$$ 
$$= \lim_{h	o 0} \frac{-6}{\sqrt{x}\sqrt{x+h}( \sqrt{x}+\sqrt{x+h})}$$ 
$$= \frac{-6}{\sqrt{x}(\sqrt{x})^2}$$ 
$$= \frac{-6}{x^{3/2}}$$ 
$$= -\frac{6}{x^2\sqrt{x}}$$

Therefore, the correct choice is D.

Find the derivative of the function f(x)=6x2 by the limit process. \text { Find the derivative of the function } f ( x ) = \frac { 6 } { \sqrt [ 2 ] { x } } \text { by the limit process. } Find the derivative of the function f(x)=2x​6​ by the limit process.

$\text { Find the derivative of the function } f ( x ) = \frac { 6 } { \sqrt [ 2 ] { x } } \text { by the limit process. }$