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

Question

Quizplus · Accepted Answer

To find the derivative of $f(t) = (1 + 8t)^{\frac{5}{9}}$, we use the chain rule. The derivative of $u^{\frac{5}{9}}$ with respect to $u$ is $\frac{5}{9}u^{\frac{5}{9}-1} = \frac{5}{9}u^{\frac{-4}{9}}$, and the derivative of $1+8t$ with respect to $t$ is $8$. Applying the chain rule, we get $f'(t) = \frac{5}{9} \cdot 8(1 + 8t)^{\frac{-4}{9}} = \frac{40}{9}(1 + 8t)^{\frac{-4}{9}}$.

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 $f ( t ) = ( 1 + 8 t ) ^ { \frac { 5 } { 9 } }$