(d)/(dx)(x^(3)sin(x))=

Apply product rule: Use the product rule for differentiation, which states that (uv)' = u'v + uv', where u = x^3 and v = sin(x).Differentiate u: Differentiate u = x^3 to get u' = 3x^2.Differentiate v: Differentiate v = sin(x) to get v' = cos(x).Apply product rule: Now apply the product rule: (x^3sin(x))' = (x^3)'sin(x) + x^3(sin(x))'.Substitute derivatives: Substitute the derivatives u' and v' into the equation: (x^3sin(x))' = 3x^2sin(x) + x^3cos(x).Final answer: So the final answer is 3x^2sin(x) + x^3cos(x).

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$\frac{d}{d x}\left(x^{3} \sin (x)\right)=$

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Algebra 2

Simplify variable expressions using properties

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\frac{d}{d x}\left(x^{3} \sin (x)\right)=

Apply product rule: Use the product rule for differentiation, which states that $(uv)' = u'v + uv'$ , where $u = x^3$ and $v = \sin(x)$ .
Differentiate $u$ : Differentiate $u = x^3$ to get $u' = 3x^2$ .
Differentiate $v$ : Differentiate $v = \sin(x)$ to get $v' = \cos(x)$ .
Apply product rule: Now apply the product rule: $x^3\sin(x)$ ' = $x^3$ '\sin(x) + x^ $3$ $\sin(x)$ '.
Substitute derivatives: Substitute the derivatives $u'$ and $v'$ into the equation: $(x^3\sin(x))' = 3x^2\sin(x) + x^3\cos(x)$ .
Final answer: So the final answer is $3x^2\sin(x) + x^3\cos(x)$ .