Find the derivative of f(x). 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 e^x with respect to x is e^x.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: f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = (1)(e^x) + (x)(e^x)Simplify Expression: Simplify the expression to get the final derivative: f'(x) = e^x + xe^x

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Find derivatives of exponential functions

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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: $\newline$ $f'(x) = e^x + xe^x$