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

Apply Quotient Rule: To find the derivative of the function y = (e^x) / x, we will use the quotient rule, which states that if you have a function that is the quotient of two functions, u(x) / v(x), then its derivative is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2.Find u(x) and v(x): Let u(x) = e^x and v(x) = x. We need to find the derivatives of u(x) and v(x). The derivative of u(x) with respect to x is u'(x) = e^x, since the derivative of e^x is e^x. The derivative of v(x) with respect to x is v'(x) = 1, since the derivative of x with respect to x is 1.Use Quotient Rule: Now we apply the quotient rule: (dy)/(dx) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Substituting u(x), u'(x), v(x), and v'(x) into this formula, we get (dy)/(dx) = (x * e^x - e^x * 1) / x^2.Simplify Expression: Simplify the expression: (dy)/(dx) = (x * e^x - e^x) / x^2. We can factor out e^x from the numerator to get (dy)/(dx) = e^x * (x - 1) / x^2.Final Derivative: The final simplified derivative of the function y = (e^x) / x is (dy)/(dx) = (e^x * (x - 1)) / x^2. This corresponds to answer choice (D).

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

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

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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)

e^{x}

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Apply Quotient Rule: To find the derivative of the function $y = \frac{e^x}{x}$ , we will use the quotient rule, which states that if you have a function that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , then its derivative is given by $\frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$ .
Find $u(x)$ and $v(x)$ : Let $u(x) = e^x$ and $v(x) = x$ . We need to find the derivatives of $u(x)$ and $v(x)$ . The derivative of $u(x)$ with respect to $x$ is $u'(x) = e^x$ , since the derivative of $e^x$ is $e^x$ . The derivative of $v(x)$ with respect to $x$ is $v(x)$ $3$ , since the derivative of $x$ with respect to $x$ is $v(x)$ $6$ .
Use Quotient Rule: Now we apply the quotient rule: $(dy)/(dx) = (v(x) \cdot u'(x) - u(x) \cdot v'(x)) / (v(x))^2$ . Substituting $u(x)$ , $u'(x)$ , $v(x)$ , and $v'(x)$ into this formula, we get $(dy)/(dx) = (x \cdot e^x - e^x \cdot 1) / x^2$ .
Simplify Expression: Simplify the expression: $(\frac{dy}{dx}) = \frac{x \cdot e^x - e^x}{x^2}$ . We can factor out $e^x$ from the numerator to get $(\frac{dy}{dx}) = \frac{e^x \cdot (x - 1)}{x^2}$ .
Final Derivative: The final simplified derivative of the function $y = \frac{e^x}{x}$ is $\frac{dy}{dx} = \frac{e^x \cdot (x - 1)}{x^2}$ . This corresponds to answer choice (D).