find \\( \\frac { d y } { d x } \\)\n\\( y = x ^ { 2 } \\cos x \\)\n\\( \\frac { d y } { d x } = \\)
find \\( \\frac { d y } { d x } \\)\n\\( y = x ^ { 2 } \\cos x \\)\n\\( \\frac { d y } { d x } = \\)
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = u\cdot v), then (\frac{dy}{dx}=u'\cdot v + u\cdot v'). Let (u = x^{2}) and (v=\cos x). First, find (u') and (v'). For (u = x^{2}), using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), we have (u'=\frac{d}{dx}(x^{2}) = 2x). For (v=\cos x), using the derivative formula (\frac{d}{dx}(\cos x)=-\sin x), we have (v'=-\sin x).
Step2: Substitute into the product rule formula
Substitute (u = x^{2}), (u' = 2x), (v=\cos x), and (v'=-\sin x) into (\frac{dy}{dx}=u'\cdot v+u\cdot v'). We get (\frac{dy}{dx}=2x\cdot\cos x+x^{2}\cdot(-\sin x)).
Step3: Simplify the expression
(\frac{dy}{dx}=2x\cos x - x^{2}\sin x).
Answer:
(2x\cos x - x^{2}\sin x)