Derivative of arcsin x

Derivative of arcsin x

Derivative $f’$ of function $f(x)=\arcsin{x}$ is:

$$ \forall x \in ]–1, 1[ ,\quad f’(x) = \frac{1}{\sqrt{1-x^2}} $$

Proof

Remember that function $\arcsin$ is the inverse function of $\sin$ :

$$ \left(f^{-1} \circ f\right)=\left(\sin \circ \arcsin\right)(x)=\sin(\arcsin(x))=x $$

Using result of derivative of inverse functions, we have:

$$ (g^{-1})^{\prime}(x)=\frac{1}{g^{\prime}(g^{-1}(x))} $$

Taking : $g^{-1}=f=\arcsin$ and $g=f^{-1}=\sin$, we have:

$$ f^{\prime}(x)=\frac{1}{\sin^{\prime}(f(x))} $$

Since:

Derivative of sin x, $g(x)=\sin x$ is:

$$ \forall x \in \mathbb{R}, \quad g’(x) = \cos x $$

So, we have:

$$ \begin{aligned} f^{\prime}(x)&=\frac{1}{\sin^{\prime}(f(x))}\\ &=\frac{1}{\cos (f(x))}\\ &=\frac{1}{\cos (\arcsin x)}\\ \end{aligned} $$

We have

$$\forall X \in \mathbb{R}, \quad \cos^2 X + \sin^2 X =1$$

and

by definition

$$ \left(f^{-1} \circ f\right)=\left(\sin \circ \arcsin \right)(x)={\color{red}{\sin(\arcsin(x))=x}} $$

Taking $X=\arcsin x$, it gives:

$$ \begin{aligned} 1&=\cos^2 X + \sin^2 X\\ &=\cos^2 (\arcsin x) + {\color{red}{\sin^2 (\arcsin x)}}\\ &=\cos^2 (\arcsin x) + {\color{red}{x^2}}\\ \end{aligned} $$

Now, we have:

$$ \cos^2 (\arcsin x) = 1 - x^2 \Longrightarrow \cos (\arcsin x) = \pm \sqrt{1 - x^2} $$

Function $\arcsin x$ is defined for all $x \in [-1,1]$ and we have

$$ \forall x \in [-1,1], \quad \arcsin x \in [-\frac{\pi}{2},\frac{\pi}{2}] $$

since it is the inverse function of $\sin:[-\frac{\pi}{2},\frac{\pi}{2}] \to [-1,1]$.

Since angle $\arcsin x \in \displaystyle [-\frac{\pi}{2},\frac{\pi}{2}]$, then cosine of this angle $\cos (\arcsin x)$ is greater than or equal to zero. Then the only possible solution is :

$$ \cos (\arcsin x) = + \sqrt{1 - x^2} $$

We conclude that:

$$ \forall x \in ]–1, 1[ ,\quad f’(x) = \frac{1}{\cos (\arcsin x)}=\frac{1}{\sqrt{1 - x^2}} $$