find the derivative of ( f(x) ).\n\n( f(x)=cos left(x^{4}\right) )\n\n( 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}))