let ( f(x)=sin left(x^{2}\right) ).\n\n( f^{prime}(x)= )\n( f^{prime}(2)= )
let ( f(x)=sin left(x^{2}\right) ).\n\n( f^{prime}(x)= )\n( f^{prime}(2)= )
Answer
Explanation:
Step1: Apply the chain rule
The chain rule states that if (y = f(g(x))), then (y'=f'(g(x))\cdot g'(x)). For (f(x)=\sin(x^{2})), let (u = x^{2}), so (f(u)=\sin(u)). The derivative of (\sin(u)) with respect to (u) is (\cos(u)), and the derivative of (u = x^{2}) with respect to (x) is (2x). Then (f'(x)=\cos(x^{2})\cdot2x=2x\cos(x^{2}))
Step2: Evaluate (f'(x)) at (x = 2)
Substitute (x = 2) into (f'(x)). We get (f'(2)=2\times2\cos(2^{2})=4\cos(4))
Answer:
(f'(x)=2x\cos(x^{2})) (f'(2)=4\cos(4))