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In one of his thousands of videos, Khan used to prove the derivative of the inverse cosine function. But I don't understand one of his steps.

Basically, he wants to find $\frac{d}{dx}\cos^{-1}x$ so he do the sub: $y=\cos^{-1}x$ then with some manipulation he arrives at: $x=\cos y$ then he takes the derivative of both sides with respect to x and he come up $1=(-\sin y)\frac{dy}{dx}$. That's the step I don't understand.

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Because he is differentiating $x = \cos y $ with respect to $x$. In other words,

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

By the chain Rule since $y$ depends on $x$

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