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

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Find derivatives using the chain rule I

Full solution

Q. Find the derivative of

\newline

f(x)

.

\newline

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

Identify Function: Identify the function that needs differentiation. The function is $f(x) = \frac{e^x}{x}$ , which is a quotient of two functions.
Use Quotient Rule: Recognize that we need to use the quotient rule for differentiation because we have a function in the form of $\frac{g(x)}{h(x)}$ , where $g(x) = e^x$ and $h(x) = x$ .
Recall Quotient Rule: Recall the quotient rule: $(\frac{d}{dx})[\frac{g(x)}{h(x)}] = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$ . We will apply this rule to find the derivative of $f(x)$ .
Differentiate Numerator: Differentiate the numerator $g(x) = e^x$ . The derivative of $e^x$ with respect to $x$ is $e^x$ .
Differentiate Denominator: Differentiate the denominator $h(x) = x$ . The derivative of $x$ with respect to $x$ is $1$ .
Apply Quotient Rule: Apply the quotient rule using the derivatives from the previous steps. The derivative of $f(x)$ is $f'(x) = \frac{e^x \cdot 1 - e^x \cdot 1}{x^2}$ .
Simplify Expression: Simplify the expression. The derivative of $f(x)$ simplifies to $f'(x) = \frac{e^x - e^x}{x^2}$ , which further simplifies to $f'(x) = \frac{0}{x^2}$ .

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