Let f(x)=(1)/(x)e^(x). Find f^(')(x). Choose 1 answer: (A) -(1)/(x^(2))((1)/(x)+e^(x)) (B) e^(x)((1)/(x)-(1)/(x^(2))) (C) -(1)/(x^(2))e^(x) (D) (1)/(x)e^(x)

Use product rule: Use the product rule for derivatives: (d/dx)(u*v) = u'v + uv'.Find u' and v': Let u = 1/x and v = e^x. Now find the derivatives u' and v'.Apply product rule: The derivative of u = 1/x is u' = -1/x^2.Substitute into formula: The derivative of v = e^x is v' = e^x.Simplify expression: Now apply the product rule: f'(x) = u'v + uv'.Combine terms: Substitute u', u, v, and v' into the formula: f'(x) = (-1/x^2)e^x + (1/x)e^x.Factor out e^x: Simplify the expression: f'(x) = -e^x/x^2 + e^x/x.Combine the terms: f'(x) = e^x(-1/x^2 + 1/x).Factor out e^x: f'(x) = e^x(-1/x^2 + 1/x).

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Let
f(x)=(1)/(x)e^(x).
Find
f^(')(x).
Choose 1 answer:
(A)
-(1)/(x^(2))((1)/(x)+e^(x))
(B)
e^(x)((1)/(x)-(1)/(x^(2)))
(C)
-(1)/(x^(2))e^(x)
(D)
(1)/(x)e^(x)

Let $f(x)=\frac{1}{x} e^{x}$ . $\newline$ Find $f^{\prime}(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-\frac{1}{x^{2}}\left(\frac{1}{x}+e^{x}\right)$ $\newline$ (B) $e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$ $\newline$ (C) $-\frac{1}{x^{2}} e^{x}$ $\newline$ (D) $\frac{1}{x} e^{x}$

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Let

f(x)=\frac{1}{x} e^{x}

\newline

Find

f^{\prime}(x)

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Use product rule: Use the product rule for derivatives: $(\frac{d}{dx})(u\cdot v) = u'v + uv'$ .
Find $u'$ and $v'$ : Let $u = \frac{1}{x}$ and $v = e^x$ . Now find the derivatives $u'$ and $v'$ .
Apply product rule: The derivative of $u = \frac{1}{x}$ is $u' = -\frac{1}{x^2}$ .
Substitute into formula: The derivative of $v = e^x$ is $v' = e^x$ .
Simplify expression: Now apply the product rule: $f'(x) = u'v + uv'$ .
Combine terms: Substitute $u'$ , $u$ , $v$ , and $v'$ into the formula: $f'(x) = (-\frac{1}{x^2})e^x + (\frac{1}{x})e^x$ .
Factor out $e^x$ : Simplify the expression: $f'(x) = -\frac{e^x}{x^2} + \frac{e^x}{x}$ .
Factor out $e^x$ : Simplify the expression: $f'(x) = -\frac{e^x}{x^2} + \frac{e^x}{x}$ .Combine the terms: $f'(x) = e^x\left(-\frac{1}{x^2} + \frac{1}{x}\right)$ .
Factor out $e^x$ : Simplify the expression: $f'(x) = -\frac{e^x}{x^2} + \frac{e^x}{x}$ .Combine the terms: $f'(x) = e^x\left(-\frac{1}{x^2} + \frac{1}{x}\right)$ .Factor out $e^x$ : $f'(x) = e^x\left(-\frac{1}{x^2} + \frac{1}{x}\right)$ .