Find the derivative of the function. $$f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }$$
A)$f ^ { t } ( t ) = \frac { 40 } { 9 } ( 1 + 8 t ) ^ { \frac { - 4 } { 9 } }$
B)$f ^ {t } ( t ) = \frac { 8 } { 9 } ( 1 + 8 t ) ^ { \frac { - 4 } { 9 } }$
C)$f ^ { t } ( t ) = \frac { 1 } { 9 } ( 1 + 8 t ) ^ { \frac { - 4 } { 9 } }$
D)$f ^ { t } ( t ) = \frac { 40 } { 9 } ( 1 + 8 t ) ^ { \frac { - 4 } { 5 } }$
E)$f ^ { t } ( t ) = 8 ( 1 + 8 t ) ^ { \frac { - 4 } { 9 } }$

Question

Quizplus · Accepted Answer

To find the derivative of $f(t) = (1 + 8t)^{\frac{5}{9}}$, we use the chain rule. The derivative of the outer function, raised to the power of $\frac{5}{9}$, is $\frac{5}{9}(1 + 8t)^{\frac{5}{9} - 1} = \frac{5}{9}(1 + 8t)^{\frac{-4}{9}}$. Then, we multiply by the derivative of the inner function, $8t$, which is $8$. Thus, the derivative is $\frac{5}{9} \cdot 8(1 + 8t)^{\frac{-4}{9}} = \frac{40}{9}(1 + 8t)^{\frac{-4}{9}}$.

Find the Derivative of the Function f(t)=(1+8t)59f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }f(t)=(1+8t)95​

Find the Derivative of the Function $f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }$