Take Derivatives of Inverse Trig Functions (ArcSin, ArcCos) – 
Take Derivatives of Inverse Trig Functions (ArcSin, ArcCos) – 

The following is proof of the derivative of $\arcsin$. Does anyone know why and with what axiom the $y’$ is introduced into the equation in the following step?

$\sin y = x$

$y’\cos y = 1$

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The following is proof of the derivative of $\arcsin$. Does anyone know why and with what axiom the $y’$ is introduced into the equation in the following step?

$\sin y = x$

$y’\cos y = 1$

The step involves applying the chain rule: \begin{align} \frac{d \sin (y)}{dx} &= \frac{d \sin (y)}{dy} \cdot \frac{dy}{dx} \\ &= \cos(y) \cdot y’ \, . \end{align} Applying the chain rule in this way, where you have an expression in terms of $y$ and want to differentiate with respect to $x$, is known as implicit differentiation. In general, $$\frac{df(y)}{dx} = \frac{df(y)}{dy} \cdot \frac{dy}{dx} \, .$$ In practice, implicit differentiation is easy because you just differentiate as you would normally, and then multiply by $dy/dx$. Examples: \begin{align} \frac{d(y^2)}{dx}&=2y\frac{dy}{dx} \\[4pt] \frac{d(\log(y))}{dx} &= \frac{1}{y}\frac{dy}{dx} \\[4pt] \frac{d\left((2y^3+5)^2\right)}{dx} &= 12y^2(2y^3+5)\frac{dy}{dx} \, . \end{align} Let me know if you have any questions.

$$\sin y=x\implies y’\cos y=1$$ is just the differentiation of the two members of an identity. Then from

$$\cos^2y+\sin^2y=1$$ the above equation can be written

$$y’\sqrt{1-\sin^2y}=\sqrt{1-x^2}=1.$$

The answer to your question is already written explicitly in the proof: “We next take the derivative of both sides of the equation …”.
Apply the chain rule to the left-hand side of the equation $\sin(y)=x$.
$\$

Your $y’=\frac{1}{\cos(y)}$ comes also from the inverse rule of differentiation $[f^{-1}]'(x)=\frac{1}{f'(f^{-1}(x)}$, from the Inverse function theorem:

Set $f=\sin$, $f^{-1}=\text{arcscin}$, $y=f^{-1}(x)$.
Set $f^{-1}(x)=\text{arcsin(x)}$ on the left-hand side of the inverse rule and $f^{-1}(x)=y$ on the right-hand side and procced as in the proof above.

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