Let y=(e^(x))/(x). Find (dy)/(dx). Choose 1 answer: (A) (e^(x-1))/(x^(2)) (B) (e^(x)(x-1))/(x^(2)) (C) e^(x) (D) (e^(x)-1)/(x^(2))

Identify function: Identify the function to differentiate. We are given the function y = (e^x) / x, and we need to find its derivative with respect to x. This is a quotient of two functions, so we will use the quotient rule for differentiation.Apply quotient rule: Apply the quotient rule. The quotient rule states that the derivative of a function h(x) = f(x) / g(x) is given by h'(x) = (g(x)f'(x) - f(x)g'(x)) / (g(x))^2. Here, f(x) = e^x and g(x) = x.Differentiate functions: Differentiate f(x) = e^x and g(x) = x. The derivative of f(x) = e^x with respect to x is f'(x) = e^x. The derivative of g(x) = x with respect to x is g'(x) = 1.Plug derivatives into formula: Plug the derivatives into the quotient rule formula. Using the derivatives from Step 3, we get h'(x) = (x * e^x - e^x * 1) / x^2.Simplify expression: Simplify the expression. Simplify the numerator: x * e^x - e^x = e^x(x - 1). Now the derivative is h'(x) = e^x(x - 1) / x^2.Match with options: Match the result with the given options. The simplified derivative e^x(x - 1) / x^2 matches option (B).

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Let
y=(e^(x))/(x).
Find
(dy)/(dx).
Choose 1 answer:
(A)
(e^(x-1))/(x^(2))
(B)
(e^(x)(x-1))/(x^(2))
(C)
e^(x)
(D)
(e^(x)-1)/(x^(2))

Let $y=\frac{e^{x}}{x}$ . $\newline$ Find $\frac{d y}{d x}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{e^{x-1}}{x^{2}}$ $\newline$ (B) $\frac{e^{x}(x-1)}{x^{2}}$ $\newline$ (C) $e^{x}$ $\newline$ (D) $\frac{e^{x}-1}{x^{2}}$

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Let

y=\frac{e^{x}}{x}

\newline

Find

\frac{d y}{d x}

\newline

Choose

1

answer:

\newline

(A)

\frac{e^{x-1}}{x^{2}}

\newline

(B)

\frac{e^{x}(x-1)}{x^{2}}

\newline

(C)

e^{x}

\newline

(D)

\frac{e^{x}-1}{x^{2}}

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y = \frac{e^x}{x}$ , and we need to find its derivative with respect to $x$ . This is a quotient of two functions, so we will use the quotient rule for differentiation.
Apply quotient rule: Apply the quotient rule. $\newline$ The quotient rule states that the derivative of a function $h(x) = \frac{f(x)}{g(x)}$ is given by $h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$ . Here, $f(x) = e^x$ and $g(x) = x$ .
Differentiate functions: Differentiate $f(x) = e^x$ and $g(x) = x$ . The derivative of $f(x) = e^x$ with respect to $x$ is $f'(x) = e^x$ . The derivative of $g(x) = x$ with respect to $x$ is $g'(x) = 1$ .
Plug derivatives into formula: Plug the derivatives into the quotient rule formula. Using the derivatives from Step $3$ , we get $h'(x) = \frac{x \cdot e^x - e^x \cdot 1}{x^2}$ .
Simplify expression: Simplify the expression. $\newline$ Simplify the numerator: $x \cdot e^x - e^x = e^x(x - 1)$ . $\newline$ Now the derivative is $h'(x) = \frac{e^x(x - 1)}{x^2}$ .
Match with options: Match the result with the given options. $\newline$ The simplified derivative $e^x(x - 1) / x^2$ matches option $(B)$ .