Identify Functions: Let's identify the two functions that we are dealing with in the product. The first function is x, and the second function is e^x. We will need to use the product rule to find the derivative of f(x) = xe^x, which 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.Derivative of x: The derivative of the first function, which is x, is 1.Derivative of e^x: The derivative of the second function, which is e^x, is e^x because the derivative of e^x with respect to x is e^x.Apply Product Rule: Now we apply the product rule. According to the product rule, the derivative of f(x) = xe^x is f'(x) = (derivative of x) * (e^x) + (x) * (derivative of e^x). Substituting the derivatives we found in the previous steps, we get f'(x) = 1 * e^x + x * e^x.Simplify Expression: Simplify the expression by combining like terms. Since both terms have a factor of e^x, we can factor it out to get f'(x) = e^x(1 + x).

Calculus

Find derivatives using the quotient rule I

Full solution

f(x)=xe^{x}

\newline

f'(x) =?

Identify Functions: Let's identify the two functions that we are dealing with in the product. The first function is $x$ , and the second function is $e^x$ . We will need to use the product rule to find the derivative of $f(x) = xe^x$ , which 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.
Derivative of $x$ : The derivative of the first function, which is $x$ , is $1$ .
Derivative of $e^x$ : The derivative of the second function, which is $e^x$ , is $e^x$ because the derivative of $e^x$ with respect to $x$ is $e^x$ .
Apply Product Rule: Now we apply the product rule. According to the product rule, the derivative of $f(x) = xe^x$ is $f'(x) = (\text{derivative of } x) \cdot (e^x) + (x) \cdot (\text{derivative of } e^x)$ . Substituting the derivatives we found in the previous steps, we get $f'(x) = 1 \cdot e^x + x \cdot e^x$ .
Simplify Expression: Simplify the expression by combining like terms. Since both terms have a factor of $e^x$ , we can factor it out to get $f'(x) = e^x(1 + x)$ .