Question

Find dy/dx when y = Ln (sinh 2x). A. 2 cosh 2x B. 2 sech 2x C. 2 coth 2x D. 2 csch 2x

Answer 1

The function given is a composite function. Let's work from the outside in:
$y=(ln(sinh(2x)))$

First step:
$\frac{d}{dx}[y]= \frac{d}{dx}[ln(sinh(2x))]$

Now, let's work it out:
$\frac{dy}{dx} = \frac{1}{sinh(2x) } * \frac{d}{dx}[sinh(2x)]$

Next step:
$\frac{dy}{dx} = \frac{1}{sinh(2x) } * cosh(2x) * \frac{d}{dx}[2x]$

Next step:
$\frac{dy}{dx} = \frac{1}{sinh(2x) } * cosh(2x) *2$

Simplify:
$\frac{dy}{dx} = \frac{2cosh(2x)}{sinh(2x) }$

Simplify further:
$\frac{dy}{dx} = 2 (\frac{cosh(2x)}{sinh(2x)})$

Remember that:
${\frac{cosh(x)}{sinh(x)} = coth(x)$

So, your final answer is:
$\boxed{ \frac{dy}{dx} = 2coth(2x) }$

So, your answer is C. Hope I could help you!