find \\( \\frac { d y } { d x } \\).\n\n\\( y = x ^ { 9 } \\cos x \\)\n\n\\( \\frac { d y } { d x } = \\)
find \\( \\frac { d y } { d x } \\).\n\n\\( y = x ^ { 9 } \\cos x \\)\n\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'v + uv'). Let (u = x^{9}) and (v=\cos x). First, find (u'): Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), so (u'=\frac{d}{dx}(x^{9})=9x^{8}). Second, find (v'): Using the derivative of cosine function (\frac{d}{dx}(\cos x)=-\sin x).
Step2: Substitute into the product rule formula
(\frac{dy}{dx}=u'v+uv'=9x^{8}\cos x+x^{9}(-\sin x))
Answer:
(9x^{8}\cos x - x^{9}\sin x)