Second derivative using implicit differentiation with respect to $x$ of $x = \sin y + \cos y$

Question

I am running into trouble with this question:

I get as far as

$$1 = \cos y\frac{dy}{dx} - \sin y\frac{dy}{dx}$$

$$1 = \frac{dy}{dx} (\cos y - \sin y)$$

$$\frac{dy}{dx} = \frac{1}{\cos y-\sin y}$$

Second derivative:

Unsure how to continue here

Answer 1

You have

$$\frac{dy}{dx}=\frac{1}{\cos y-\sin y}.$$

Take the derivative of both sides using one of the derivative rules:

$$\frac{d^2y}{dx^2}=\frac{(1)'(\cos y - \sin y)-(1)(\cos y-\sin y)'}{(\cos y-\sin y)^2}=?,$$

$$\mathrm{or},\quad (x^{-1})'=-x^{-2}\implies\frac{d^2y}{dx^2}=\frac{-1}{(\cos y-\sin y)^2}\cdot(\cos y-\sin y)'=?$$

Above are the beginnings to (i) the quotient rule and (ii) the power rule and chain rules.

Answer 2

Just differentiate both sides of ${dy\over dx}=(\cos y-\sin y)^{-1}$ with respect to $x$. This leads to: $${d^2y\over dx^2}={d\over dx}{dy\over dx}={d\over dx}(\cos y-\sin y)^{-1}=-(\cos y-\sin y)^{-2}\cdot\Bigl[ (-\sin y){dy\over dx} -(\cos y){dy\over dx}\Bigr].$$