Question

hw11 differentiation rules ii (target c1, c2, c5; §3.3)
score: 0/8 answered: 0/8
question 1
(1) which is the correct formula for finding the derivative of the product of two functions?
○(fg) = fg
○(fg) = fg+gf
○(fg) = f+g
(2) use the correct formula above to find the derivative of the function f(x)=x^4e^x.
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Explanation:

Step1: Recall product - rule formula

The product - rule for the derivative of the product of two functions \(f(x)\) and \(g(x)\) is \((fg)'=fg'+gf'\).

Step2: Identify functions for \(f(x)=x^{4}e^{x}\)

Let \(u = x^{4}\) and \(v=e^{x}\). Then \(u'=\frac{d}{dx}(x^{4}) = 4x^{3}\) and \(v'=\frac{d}{dx}(e^{x})=e^{x}\).

Step3: Apply product - rule

Using the product - rule \((uv)'=uv'+vu'\), we substitute \(u = x^{4}\), \(u' = 4x^{3}\), \(v = e^{x}\), and \(v'=e^{x}\). So \(f'(x)=x^{4}e^{x}+4x^{3}e^{x}=x^{3}e^{x}(x + 4)\).

Answer:

(1) \((fg)'=fg'+gf'\)
(2) \(x^{3}e^{x}(x + 4)\)