Implicit differentiation of $\log$ and $\sin$ function

Question

I am struggling with the following problem for implicit differentiation.

I am tasked to differentiate implicitly the following function, and evaluate $y''(0)$, where $y=y(x)$.

$$\ln(y+1)+\sin(xy)=\ln(5).$$

I have differentiated this once to find,

$$(y+xy')\cos(xy)+\frac{y'}{y+1}=0$$

But how to advance from here to find $y''(0)$?

Thanks

Answer 1

In the original equation we can find $y(0)$:

$$\ln(y+1) + \sin 0 = \ln 5 \implies y(0) = 4$$

And in that equation we can find $y'(0)$:

$$(4+0)\cos 0 + \frac{y'}{5} = 0 \implies y'(0) = -20$$

Now just implicit differentiate that expression again and follow the same procedure. Can you take it from here?