if ( f(x)=arccos(x^{2}) ), then ( f(x)= )

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = f(g(x))), then (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). Let (u = x^{2}), so (f(x)=\arccos(u)). The derivative of (\arccos(u)) with respect to (u) is (-\frac{1}{\sqrt{1 - u^{2}}}), and the derivative of (u=x^{2}) with respect to (x) is (2x).

Step2: Substitute back (u = x^{2})

Using the chain - rule formula (f^\prime(x)=-\frac{1}{\sqrt{1-(x^{2})^{2}}}\cdot2x)

Answer:

(f^\prime(x)=-\frac{2x}{\sqrt{1 - x^{4}}})