(d)/(dx)(e^(x)cos(x))=

Apply product rule: Use the product rule for differentiation, which states that (d/dx)(u*v) = u'v + uv'. Here, u = e^x and v = cos(x).Differentiate u: Differentiate u = e^x. The derivative of e^x with respect to x is e^x. So, u' = e^x.Differentiate v: Differentiate v = cos(x). The derivative of cos(x) with respect to x is -sin(x). So, v' = -sin(x).Apply product rule: Apply the product rule: (d/dx)(e^x * cos(x)) = e^x * (-sin(x)) + e^x * cos(x).Simplify expression: Simplify the expression: e^x * (-sin(x)) + e^x * cos(x) = -e^xsin(x) + e^xcos(x).Combine like terms: Combine like terms, if any. But there are no like terms to combine here.

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$\frac{d}{d x}\left(e^{x} \cos (x)\right)=$

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Algebra 2

Csc, sec, and cot of special angles

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\frac{d}{d x}\left(e^{x} \cos (x)\right)=

Apply product rule: Use the product rule for differentiation, which states that $(\frac{d}{dx})(u*v) = u'v + uv'$ . Here, $u = e^x$ and $v = \cos(x)$ .
Differentiate $u$ : Differentiate $u = e^x$ . The derivative of $e^x$ with respect to $x$ is $e^x$ . $\newline$ So, $u' = e^x$ .
Differentiate $v$ : Differentiate $v = \cos(x)$ . The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$ . $\newline$ So, $v' = -\sin(x)$ .
Apply product rule: Apply the product rule: $(\frac{d}{dx})(e^x \cdot \cos(x)) = e^x \cdot (-\sin(x)) + e^x \cdot \cos(x)$ .
Simplify expression: Simplify the expression: $e^x \cdot (-\sin(x)) + e^x \cdot \cos(x) = -e^x\sin(x) + e^x\cos(x)$ .
Combine like terms: Combine like terms, if any. But there are no like terms to combine here.