Question

find the derivative of ( f(x) ).

( f(x)=cos left(x^{4}
ight) )

( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(g(x))\), then \(y'=f'(g(x))\cdot g'(x)\). Let \(u = x^{4}\), so \(f(x)=\cos(u)\). The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\), i.e., \(f'(u)=-\sin(u)\).

Step2: Find the derivative of the inner - function

The derivative of \(u = x^{4}\) with respect to \(x\) is \(u'=\frac{d}{dx}(x^{4}) = 4x^{3}\) using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\).

Step3: Combine using the chain rule

Substitute \(u = x^{4}\) and its derivative into the chain - rule formula. \(f'(x)=-\sin(x^{4})\cdot4x^{3}\)

Answer:

\(- 4x^{3}\sin(x^{4})\)