Question

if $f(x)=arccos(x^{2})$, then $f(x)=$
(a) $\frac{1}{sqrt{1 - x^{4}}}$
(b) $\frac{-2x}{sqrt{1 - x^{4}}}$
(c) $\frac{2x}{sqrt{1 - x^{4}}}$
(d) $\frac{-4x^{3}}{sqrt{1 - x^{4}}}$
(e) $\frac{4x^{3}}{sqrt{1 - x^{4}}}$

Explanation:

Step1: Recall derivative formula

The derivative of $y = \arccos(u)$ is $y'=-\frac{1}{\sqrt{1 - u^{2}}}\cdot u'$. Here $u = x^{2}$.

Step2: Differentiate $u$

The derivative of $u=x^{2}$ with respect to $x$ is $u' = 2x$.

Step3: Apply chain - rule

Substitute $u = x^{2}$ and $u'=2x$ into the derivative formula for $\arccos(u)$. We get $f'(x)=-\frac{1}{\sqrt{1-(x^{2})^{2}}}\cdot2x=-\frac{2x}{\sqrt{1 - x^{4}}}$.

Answer:

B. $\frac{-2x}{\sqrt{1 - x^{4}}}$