Let y=cos(x)x^(3). (dy)/(dx)=

Apply Product Rule: Use the product rule for differentiation, which states that (d/dx)[u*v] = u'v + uv', where u = cos(x) and v = x^(3). Differentiate u = cos(x): First, differentiate u = cos(x) with respect to x to get u' = -sin(x). Differentiate v = x^(3): Then, differentiate v = x^(3) with respect to x to get v' = 3x^(2). Use Product Rule Formula: Now apply the product rule: (dy)/(dx) = u'v + uv'. Substitute into Formula: Substitute u', u, v', and v into the formula: (dy)/(dx) = (-sin(x))(x^(3)) + (cos(x))(3x^(2)). Simplify Expression: Simplify the expression: (dy)/(dx) = -x^(3)sin(x) + 3x^(2)cos(x).

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Algebra 2

Simplify variable expressions using properties

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Q. Let

y=\cos (x) x^{3}

\newline

\frac{d y}{d x}=

Apply Product Rule: Use the product rule for differentiation, which states that $\frac{d}{dx}[u*v] = u'v + uv'$ , where $u = \cos(x)$ and $v = x^{3}$ .
Differentiate $u = \cos(x)$ : First, differentiate $u = \cos(x)$ with respect to $x$ to get $u' = -\sin(x)$ .
Differentiate $v = x^{3}$ : Then, differentiate $v = x^{3}$ with respect to $x$ to get $v' = 3x^{2}$ .
Use Product Rule Formula: Now apply the product rule: $(\frac{dy}{dx}) = u'v + uv'$ .
Substitute into Formula: Substitute $u'$ , $u$ , $v'$ , and $v$ into the formula: $\frac{dy}{dx} = (-\sin(x))(x^{3}) + (\cos(x))(3x^{2})$ .
Simplify Expression: Simplify the expression: $\frac{dy}{dx} = -x^{3}\sin(x) + 3x^{2}\cos(x)$ .