Find the derivative of y with respect to the appropriate variable.
-$$y = ( 1 - 7 t ) \operatorname { coth } ^ { - 1 } \sqrt { 7 t }$$
A) $\frac { \sqrt { 7 } } { 2 \sqrt { t } } - 7 \operatorname { coth } ^ { - 1 } \sqrt { 7 t }$
B) $- \frac { 7 } { 2 } t$
C) $( 1 - 7 t ) \operatorname { coth } ^ { - 1 } \sqrt { 7 t }$
D) $\frac { \sqrt { 7 } } { 2 \sqrt { t } } - 7 \tanh ^ { - 1 } \sqrt { 7 t }$

Question

Quizplus · Accepted Answer

To find the derivative of $y = ( 1 - 7 t ) \operatorname { coth } ^ { - 1 } \sqrt { 7 t }$, we apply the product rule and the chain rule. The derivative of $1 - 7t$ is $-7$, and the derivative of $\operatorname { coth } ^ { - 1 } \sqrt { 7 t }$ involves applying the chain rule twice: first for the inverse hyperbolic cotangent function and then for the square root function. The derivative of $\operatorname { coth } ^ { - 1 } x$ is $-\frac{1}{1-x^2}$, and for $\sqrt{7t}$, it's $\frac{1}{2\sqrt{7t}} \cdot 7$. Combining these with the product rule gives the correct answer.

Find the Derivative of Y with Respect to the Appropriate y=(1−7t)coth⁡−17ty = ( 1 - 7 t ) \operatorname { coth } ^ { - 1 } \sqrt { 7 t }y=(1−7t)coth−17t​

Find the Derivative of Y with Respect to the Appropriate $y = ( 1 - 7 t ) \operatorname { coth } ^ { - 1 } \sqrt { 7 t }$