Question

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:

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