Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
-$$y = 4 x ^ { x ^ { 2 } }$$
A) $4 x ^ { x ^ { 2 } + 1 } ( 2 \ln 4 x )$
B) $x ( 2 \ln 4 x + 1 )$
C) $4 x ^ { x ^ { 2 } + 1 } ( 2 \ln 4 x + 1 )$
D) $x ^ { x ^ { 2 } + 1 } \ln 4 x$

Question

Quizplus · Accepted Answer

Using logarithmic differentiation, we have
\begin{align*}
\ln y &= \ln ( 4 x ^ { x ^ { 2 } } ) \
&= \ln 4 + \ln ( x ^ { x ^ { 2 } } ) \
&= \ln 4 + x ^ { 2 } \ln x \
\frac{1}{y} \cdot y' &= \frac{d}{dx} ( \ln 4 + x ^ { 2 } \ln x ) \
&= \frac{1}{x} + 2x \ln x \
y' &= y \cdot \left( \frac{1}{x} + 2x \ln x ight) \
&= 4 x ^ { x ^ { 2 } } \cdot \left( \frac{1}{x} + 2x \ln x ight) \
&= 4x^{x^2+1}(2\ln 4x+1)
\end{align*}
Therefore, the correct choice is C.

Use Logarithmic Differentiation to Find the Derivative of Y with Respect