$\text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. }$
A)
$f ^ { tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { 4 } { 9 } }$
B)
$f ^ {tt } ( x ) = \frac { 5 } { 81 } x ^ { \frac { - 13 } { 9 } }$
C)
$f ^ {tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$
D)
$f ^ { tt } ( x ) = \frac { 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$
E)
$f ^ { tt} ( x ) = 8 x ^ { \frac { - 13 } { 9 } }$

Question

$\text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. }$ 
A)
$f ^ { tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { 4 } { 9 } }$
B)
$f ^ {tt } ( x ) = \frac { 5 } { 81 } x ^ { \frac { - 13 } { 9 } }$
C)
$f ^ {tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$
D)
$f ^ { tt } ( x ) = \frac { 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$
E)
$f ^ { tt} ( x ) = 8 x ^ { \frac { - 13 } { 9 } }$

Quizplus · Accepted Answer

The first derivative of the given function is:
$$f'(x) = \frac{40}{9}x^{4/9}$$
Now, taking the derivative of $f'(x)$ with respect to $x$ gives the second derivative of $f(x)$:
$$f''(x) = \frac{40}{9} \cdot \frac{4}{9} x^{-5/9} = \frac{-160}{81}x^{-5/9} = \frac{-160}{81}x^{-13/9}$$
Therefore, the correct choice is (C).

Find the second derivative of the function f(x)=8x59. \text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. } Find the second derivative of the function f(x)=8x95​.

$\text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. }$