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find the derivative $\frac{d}{dx}(e^{x}\sin(x))$

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Find derivatives of exponential functions

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Q. find the derivative

\frac{d}{dx}(e^{x}\sin(x))

Identify Function & Apply Rule: Identify the function and apply the product rule. $\newline$ The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. $\newline$ Let $u = e^x$ and $v = \sin(x)$ . Then, $f(x) = u \cdot v$ . $\newline$ $f'(x) = u' \cdot v + u \cdot v'$ .
Differentiate $e^x$ : Differentiate $u = e^x$ . $\newline$ The derivative of $e^x$ with respect to $x$ is $e^x$ . $\newline$ $u' = \frac{d}{dx}(e^x) = e^x$ .
Differentiate $\sin(x)$ : Differentiate $v = \sin(x)$ . $\newline$ The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$ . $\newline$ $v' = \frac{d}{dx}(\sin(x)) = \cos(x)$ .
Apply Derivatives to Rule: Apply the derivatives found in steps $2$ and $3$ to the product rule. $\newline$ $f'(x) = u' \cdot v + u \cdot v' = e^x \cdot \sin(x) + e^x \cdot \cos(x)$ .
Combine Like Terms: Combine like terms if possible. $\newline$ In this case, there are no like terms to combine, so the derivative remains as is. $\newline$ $f'(x) = e^x \cdot \sin(x) + e^x \cdot \cos(x)$ .

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