assignment 5\nscore: 51/57 answered: 20/22\nquestion 21\ndetails\nfind the first derivative of ( f(x)=6…

Explanation:

Step1: Apply the product rule

The product rule for three functions (u(x)), (v(x)), (w(x)) is ((uvw)'=u'vw + uv'w+uvw'). Let (u = 6x^{6}), (v=e^{x}), (w = \sin(x)). First, find (u'=\frac{d}{dx}(6x^{6})=36x^{5}), (v'=\frac{d}{dx}(e^{x})=e^{x}), (w'=\frac{d}{dx}(\sin(x))=\cos(x)).

Step2: Substitute into the product - rule formula

[ \begin{align*} f'(x)&=u'vw+uv'w + uvw'\ &=36x^{5}\cdot e^{x}\cdot\sin(x)+6x^{6}\cdot e^{x}\cdot\sin(x)+6x^{6}\cdot e^{x}\cdot\cos(x)\ &=6x^{5}e^{x}(6\sin(x)+x\sin(x)+x\cos(x)) \end{align*} ]

Answer:

(6x^{5}e^{x}(6\sin(x)+x\sin(x)+x\cos(x)))