"Find the derivative of f(x). f(x) = (e^x)/(x) f′(x) = ______"

Identify the function: Let's identify the function we are differentiating. The function is f(x) = (e^x)/(x). We will use the quotient rule to find the derivative, which states that if we have a function g(x) = u(x)/v(x), then g'(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))^2.Identify numerator and its derivative: Let's identify the numerator u(x) and its derivative u'(x). The numerator is e^x, and the derivative of e^x with respect to x is e^x.Identify denominator and its derivative: Now, let's identify the denominator v(x) and its derivative v'(x). The denominator is x, and the derivative of x with respect to x is 1.Apply the quotient rule: Applying the quotient rule, we get f′(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))^2. Substituting u(x) = e^x, u'(x) = e^x, v(x) = x, and v'(x) = 1, we get f′(x) = (e^x * x - e^x * 1) / (x^2).Simplify the expression: Simplify the expression in the numerator to get f′(x) = (e^x * (x - 1)) / (x^2).

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Calculus

Find derivatives using the quotient rule I

Full solution

Q. Find the derivative of

f(x)

\newline

f(x) = \frac{e^x}{x}

\newline

f'(x) =

______

Identify the function: Let's identify the function we are differentiating. The function is $f(x) = \frac{e^x}{x}$ . We will use the quotient rule to find the derivative, which states that if we have a function $g(x) = \frac{u(x)}{v(x)}$ , then $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ .
Identify numerator and its derivative: Let's identify the numerator $u(x)$ and its derivative $u'(x)$ . The numerator is $e^x$ , and the derivative of $e^x$ with respect to $x$ is $e^x$ .
Identify denominator and its derivative: Now, let's identify the denominator $v(x)$ and its derivative $v'(x)$ . The denominator is $x$ , and the derivative of $x$ with respect to $x$ is $1$ .
Apply the quotient rule: Applying the quotient rule, we get $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Substituting $u(x) = e^x$ , $u'(x) = e^x$ , $v(x) = x$ , and $v'(x) = 1$ , we get $f'(x) = \frac{e^x \cdot x - e^x \cdot 1}{x^2}$ .
Simplify the expression: Simplify the expression in the numerator to get $f'(x) = \frac{e^x (x - 1)}{x^2}$ .