Differentiate $y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }$ .(Do not simplify)
A) $\frac { - 4 \sinh ( x ) } { \cosh ^ { 2 } x }$
B) $\frac { - 4 \sinh ( 4 x ) \cdot \cosh ^ { 2 } ( x ) - 2 \sinh x \cdot \cosh ( 4 x ) } { ( \cosh x ) ^ { 4 } }$
C) $\frac { 4 \sinh ( 4 x ) \cdot \cosh ^ { 2 } ( x ) - 2 \sinh x \cdot \cosh ( x ) \cdot \cosh ( 4 x ) } { ( \cosh x ) ^ { 4 } }$
D) $\frac { - 4 \sinh ( 4 x ) \cdot \cosh ^ { 2 } ( x ) + 2 \sinh x \cdot \cosh ( x ) \cdot \cosh ( 4 x ) } { ( \cosh x ) ^ { 4 } }$

Question

Quizplus · Accepted Answer

The derivative is found using the quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$, where $u = \cosh(4x)$ and $v = \cosh^2(x)$. Differentiating, $u' = 4\sinh(4x)$ and $v' = 2\sinh(x)\cosh(x)$. Substituting these into the quotient rule gives the expression in choice C.

Differentiate y=cosh⁡(4x)cosh⁡2(x)y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }y=cosh2(x)cosh(4x)​

Differentiate $y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }$