Question

find the derivative of the function below.
$f(x)=arcsin(sqrt{1 - x^{6}})$

Explanation:

Step1: Apply chain - rule

Let $u = \sqrt{1 - x^{6}}$, then $f(x)=\arcsin(u)$. The derivative of $\arcsin(u)$ with respect to $u$ is $\frac{1}{\sqrt{1 - u^{2}}}$, and we need to multiply by the derivative of $u$ with respect to $x$.

Step2: Find derivative of $u$

$u=(1 - x^{6})^{\frac{1}{2}}$. Using the chain - rule again, let $v = 1 - x^{6}$, then $u = v^{\frac{1}{2}}$. The derivative of $u$ with respect to $v$ is $\frac{1}{2}v^{-\frac{1}{2}}$, and the derivative of $v$ with respect to $x$ is $-6x^{5}$. So, $\frac{du}{dx}=\frac{1}{2}(1 - x^{6})^{-\frac{1}{2}}\times(-6x^{5})=\frac{-3x^{5}}{\sqrt{1 - x^{6}}}$.

Step3: Combine derivatives

Since $\frac{df}{du}=\frac{1}{\sqrt{1 - u^{2}}}$ and $u = \sqrt{1 - x^{6}}$, then $\frac{df}{du}=\frac{1}{\sqrt{1-(1 - x^{6})}}=\frac{1}{\sqrt{x^{6}}}=\frac{1}{|x^{3}|}$ (for $x
eq0$). And $\frac{df}{dx}=\frac{df}{du}\times\frac{du}{dx}$. So, $\frac{df}{dx}=\frac{1}{|x^{3}|}\times\frac{-3x^{5}}{\sqrt{1 - x^{6}}}=\frac{-3x^{2}}{\sqrt{1 - x^{6}}}$ for $|x|\lt1$ and $x
eq0$.

Answer:

$\frac{-3x^{2}}{\sqrt{1 - x^{6}}},|x|\lt1,x
eq0$