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Find the derivative of $\newline$ $f(x)$ . $\newline$ $f(x)=xe^{x}$

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Find derivatives of exponential functions

Full solution

Q. Find the derivative of

\newline

f(x)

.

\newline

f(x)=xe^{x}

Apply Product Rule: To find the derivative of the function $f(x) = xe^x$ , we need to apply the product rule because the function is the product of two functions, $x$ and $e^x$ . The product rule states that if we have two functions $u(x)$ and $v(x)$ , then the derivative of their product $u(x)v(x)$ is given by $u'(x)v(x) + u(x)v'(x)$ .
Identify Functions: Let's identify the two functions we are working with: $u(x) = x$ and $v(x) = e^x$ . We need to find the derivatives of both functions. The derivative of $u(x)$ with respect to $x$ is $u'(x) = 1$ , since the derivative of $x$ is $1$ . The derivative of $v(x)$ with respect to $x$ is $v'(x) = e^x$ , since the derivative of $v(x) = e^x$ $0$ with respect to $x$ is $v(x) = e^x$ $0$ .
Apply Product Rule: Now we apply the product rule. We multiply the derivative of $u(x)$ by $v(x)$ and add the product of $u(x)$ and the derivative of $v(x)$ . This gives us: $\newline$ $f'(x) = u'(x)v(x) + u(x)v'(x)$ $\newline$ $f'(x) = (1)(e^x) + (x)(e^x)$
Simplify Expression: Simplify the expression to get the final derivative: $\newline$ $f'(x) = e^x + xe^x$

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