Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
-$$y = 4 t \sqrt { t }$$
A) $\frac { 4 t \sqrt { t } + 1 ( \ln 4 t ) } { \sqrt { t } + 1 }$
B) $4 t \left( \frac { 1 } { 2 } \ln 4 t + 1 \right)$
C) $\frac { 1 } { \sqrt { t } } \left( \frac { 1 } { 2 } \ln 4 t + 1 \right)$
D) $\frac { 4 \mathrm { t } \sqrt { \mathrm { t } } } { \sqrt { \mathrm { t } } } \left( \frac { 1 } { 2 } \ln 4 \mathrm { t } + 1 \right)$

Question

Quizplus · Accepted Answer

Logarithmic differentiation is not directly needed for this function. Instead, we can differentiate $y = 4t\sqrt{t}$ directly using the power rule. Rewrite $y = 4t^{3/2}$. Differentiating gives $y' = 4 \cdot \frac{3}{2}t^{1/2} = 6t^{1/2}$. This simplifies to $6\sqrt{t}$, which is equivalent to the expression in choice D when factored and simplified correctly. The other options incorrectly apply logarithmic differentiation or misapply differentiation rules.

Use Logarithmic Differentiation to Find the Derivative of Y with Respect