`f(x) = x^4 e^x` Find f'(x) and f''(x)

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Chapter 3, 3.2 - Problem 27 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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tonys538 | Student, Undergraduate | (Level 1) Valedictorian

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The function f(x) = `x^4*e^x` .

f'(x) denotes the first derivative of the function. f(x) is a product of two expressions `x^4` and` e^x` . The product rule should be used to find `f'(x)` .

`f'(x) = (x^4*e^x)'`

= `x^4*(e^x)' + (x^4)'*e^x`

= `x^4*e^x + 4*x^3*e^x`

The second derivative `f''(x)`

= `(x^4*e^x + 4*x^3*e^x)'`

= `(x^4*e^x)' + (4*x^3*e^x)'`

= `x^4*e^x + 4*x^3*e^x + 4*(x^3*e^x + 3*x^2*e^x)`

= `x^4*e^x + 4*x^3*e^x + 4*x^3*e^x + 12*x^2*e^x`

= `x^4*e^x + 8*x^3*e^x + 12*x^2*e^x`

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