Derivative of arccos x

Derivative of arccos x

Derivative $f’$ of function $f(x)=\arccos{x}$ 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 $\cos$ :

$$ \left(f^{-1} \circ f\right)=\left(\cos \circ \arccos\right)(x)=\cos(\arccos(x))=x $$

Using result of derivative of inverse functions, we have:

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

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

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

Since:

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

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

So, we have:

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

We have

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

and

by definition

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

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

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

Now, we have:

$$ \sin^2 (\arccos x) = 1 - x^2 \Longrightarrow \sin (\arccos x) = \pm \sqrt{1 - x^2} $$

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

$$ \forall x \in [-1,1], \quad \arccos x \in [0, \pi] $$

since it is the inverse function of $\cos:[0, \pi] \to [-1,1]$.

Since angle $\arccos x \in \displaystyle [0, \pi]$, then sine of this angle $\sin (\arccos x)$ is greater than or equal to zero. Then the only possible solution is :

$$ \sin (\arccos x) = + \sqrt{1 - x^2} $$

We conclude that:

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