Rewrite the expression in terms of exponentials and simplify the results.
-$\sinh ( 6 \ln x )$
A) $\frac { 1 } { 2 } \left( x ^ { 6 } + \frac { 1 } { x ^ { 6 } } \right)$
B) $2 x$
C) $3 \left( x - \frac { 1 } { x } \right)$
D) $\frac { 1 } { 2 } \left( x ^ { 6 } - \frac { 1 } { x ^ { 6 } } \right)$

Question

Quizplus · Accepted Answer

We know that $\sinh (6\ln x) = \dfrac{e^{6\ln x} - e^{-6\ln x}}{2}$. Simplifying, we get:
$$\dfrac{e^{6\ln x} - e^{-6\ln x}}{2} = \dfrac{(e^{\ln x})^6 - (e^{\ln x})^{-6}}{2} = \dfrac{x^6 - \frac{1}{x^6}}{2} = \boxed{\frac{1}{2}\left(x^6 - \frac{1}{x^6}ight)}$$

Therefore, the answer is $\boxed{	extbf{(D)}}$.

Rewrite the Expression in Terms of Exponentials and Simplify the Results