$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$ $y = x ^ { 8 x }$
A) $8 x ^ { 7 x }$
B) $8 x ^ { 8 x - 1 }$
C) $8 x ^ { 8 x } \ln x$
D) $8 ( \ln x + 1 )$
E) $8 x ^ { 8 x } ( \ln x + 1 )$

Question

Quizplus · Accepted Answer

We take the natural logarithm of both sides and then differentiate both sides with respect to x using the chain rule:

\begin{aligned} y &= x^{8x}\ \ln y &= \ln \left( x^{8x} ight)\ \ln y &= 8x \ln x\ \frac{1}{y} \cdot \frac { dy } { d x } &= 8 \ln x + 8x \cdot \frac { 1 } { x }\ \frac { d y } { d x } &= 8 x ^ { 8 x } ( \ln x + 1 ) \end{aligned}

Therefore, our answer is choice E.

Use logarithmic differentiation to find dydx. \text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. } Use logarithmic differentiation to find dxdy​.

$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$