question 22\ndetails\nfind the first derivative of ( f(x)=-9 x^{4} e^{x} cos (x) ).\n( f^{prime}(x)= )
question 22\ndetails\nfind the first derivative of ( f(x)=-9 x^{4} e^{x} cos (x) ).\n( f^{prime}(x)= )
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = u\cdot v\cdot w), then (y^\prime=u^\prime vw + uv^\prime w+uvw^\prime). Let (u=-9x^{4}), (v = e^{x}), (w=\cos(x)). (u^\prime=-36x^{3}), (v^\prime = e^{x}), (w^\prime=-\sin(x))
Step2: Substitute into the product rule formula
[ \begin{align*} f^\prime(x)&=(-36x^{3})e^{x}\cos(x)+(-9x^{4})e^{x}\cos(x)+(-9x^{4})e^{x}(-\sin(x))\ &=-36x^{3}e^{x}\cos(x)-9x^{4}e^{x}\cos(x)+9x^{4}e^{x}\sin(x)\ &=-9x^{3}e^{x}(4\cos(x)+x\cos(x)-x\sin(x)) \end{align*} ]
Answer:
(-9x^{3}e^{x}(4\cos(x)+x\cos(x)-x\sin(x)))