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Derive the differentiation formula $\dfrac{d}{d x}[\arcsin x]=\dfrac{1}{\sqrt{1-x^{2}}}$.
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To derive the differentiation formula for $\frac{d}{d x}[\arcsin x]=\frac{1}{\sqrt{1-x^2}}$, we will start by defining $y=\arcsin x$. This means that $x=\sin y$

Now, we'll differentiate both sides of the equation $x=\sin y$ with respect to $x$, applying implicit differentiation. Remember that we'll need to use the chain rule when differentiating the term involving $y$ :
$\frac{d}{d x}(x)=\frac{d}{d x}(\sin y)$
Differentiating both sides, we get:
$1=(\cos y) \frac{d y}{d x}$
Now, we need to express $\cos y$ in terms of $x$. To do this, we use the Pythagorean identity:
$\sin ^2 y+\cos ^2 y=1$
Since we know that $x=\sin y$, we can substitute $x$ for $\sin y$ :
$x^2+\cos ^2 y=1$
Now, solve for $\cos y$ :

$\cos y=\sqrt{1-x^2}$
Substitute this expression for $\cos y$ back into our original equation:
$1=\left(\sqrt{1-x^2}\right) \frac{d y}{d x}$
Now, solve for $\frac{d y}{d x}$ :
$\frac{d y}{d x}=\frac{1}{\sqrt{1-x^2}}$
Since $y=\arcsin x$, we can express the result as:
$\frac{d}{d x}[\arcsin x]=\frac{1}{\sqrt{1-x^2}}$
And we've derived the differentiation formula for the arcsine function.
by Diamond (89,043 points)

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