The function given is a composite function. Let's work from the outside in:
[tex]y=(ln(sinh(2x)))[/tex]
First step:
[tex] \frac{d}{dx}[y]= \frac{d}{dx}[ln(sinh(2x))][/tex]
Now, let's work it out:
[tex]\frac{dy}{dx} = \frac{1}{sinh(2x) } * \frac{d}{dx}[sinh(2x)][/tex]
Next step:
[tex]\frac{dy}{dx} = \frac{1}{sinh(2x) } * cosh(2x) * \frac{d}{dx}[2x] [/tex]
Next step:
[tex]\frac{dy}{dx} = \frac{1}{sinh(2x) } * cosh(2x) *2[/tex]
Simplify:
[tex]\frac{dy}{dx} = \frac{2cosh(2x)}{sinh(2x) }[/tex]
Simplify further:
[tex] \frac{dy}{dx} = 2 (\frac{cosh(2x)}{sinh(2x)})[/tex]
Remember that:
[tex] {\frac{cosh(x)}{sinh(x)} = coth(x)[/tex]
So, your final answer is:
[tex]\boxed{ \frac{dy}{dx} = 2coth(2x) }[/tex]
So, your answer is C. Hope I could help you!
