Find (d)/(dx)(e^((1)/(x))) Choose 1 answer: (A) -(e^((1)/(x)))/(x^(2)) (B) (1)/(x)*e^((1)/(x)-1) (C) -e^((1)/(x)) (D) (e^((1)/(x)))/(x)

Identify function: Identify the function to differentiate. We need to find the derivative of the function f(x) = e^(1/x) with respect to x.Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is e^u, where u = 1/x, and the inner function is 1/x.Differentiate outer function: Differentiate the outer function. The derivative of e^u with respect to u is e^u. So, the derivative of e^(1/x) with respect to 1/x is e^(1/x).Differentiate inner function: Differentiate the inner function. The derivative of 1/x with respect to x is -1/x^2.Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. The derivative of f(x) with respect to x is e^(1/x) times -1/x^2.Write final answer: Write the final answer. The derivative of e^(1/x) with respect to x is -(e^(1/x))/(x^2), which corresponds to answer choice (A).

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Find
(d)/(dx)(e^((1)/(x)))
Choose 1 answer:
(A)
-(e^((1)/(x)))/(x^(2))
(B)
(1)/(x)*e^((1)/(x)-1)
(C)
-e^((1)/(x))
(D)
(e^((1)/(x)))/(x)

Find $\frac{d}{d x}\left(e^{\frac{1}{x}}\right)$ $\newline$ Choose $1$ answer: $\newline$ (A) $-\frac{e^{\frac{1}{x}}}{x^{2}}$ $\newline$ (B) $\frac{1}{x} \cdot e^{\frac{1}{x}-1}$ $\newline$ (C) $-e^{\frac{1}{x}}$ $\newline$ (D) $\frac{e^{\frac{1}{x}}}{x}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Find

\frac{d}{d x}\left(e^{\frac{1}{x}}\right)

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Identify function: Identify the function to differentiate. $\newline$ We need to find the derivative of the function $f(x) = e^{\frac{1}{x}}$ with respect to $x$ .
Apply chain rule: Apply the chain rule for differentiation. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $e^u$ , where $u = \frac{1}{x}$ , and the inner function is $\frac{1}{x}$ .
Differentiate outer function: Differentiate the outer function. $\newline$ The derivative of $e^u$ with respect to $u$ is $e^u$ . So, the derivative of $e^{1/x}$ with respect to $1/x$ is $e^{1/x}$ .
Differentiate inner function: Differentiate the inner function. $\newline$ The derivative of $\frac{1}{x}$ with respect to $x$ is $-\frac{1}{x^2}$ .
Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ The derivative of $f(x)$ with respect to $x$ is $e^{\frac{1}{x}}$ times $-\frac{1}{x^2}$ .
Write final answer: Write the final answer. $\newline$ The derivative of $e^{1/x}$ with respect to $x$ is $-\left(\frac{e^{1/x}}{x^2}\right)$ , which corresponds to answer choice $(A)$ .