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

Product Rule Derivative: Use the product rule for derivatives: (d/dx)(u*v) = u'v + uv'.Identify u and v: Let u = 1/x and v = e^(x). Now find the derivatives u' and v'.Calculate u' and v': The derivative of u = 1/x is u' = -1/x^2.Apply Product Rule: The derivative of v = e^(x) is v' = e^(x).Substitute into Formula: Apply the product rule: f'(x) = u'v + uv'.Simplify Expression: Substitute u', u, v, and v' into the formula: f'(x) = (-1/x^2)e^(x) + (1/x)e^(x).Final Answer: Simplify the expression: f'(x) = e^(x)(-1/x^2 + 1/x).The final answer is f'(x) = e^(x)((1/x) - (1/x^2)).

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

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

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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}} e^{x}

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Product Rule Derivative: Use the product rule for derivatives: $(\frac{d}{dx})(u\cdot v) = u'v + uv'$ .
Identify $u$ and $v$ : Let $u = \frac{1}{x}$ and $v = e^{x}$ . Now find the derivatives $u'$ and $v'$ .
Calculate $u'$ and $v'$ : The derivative of $u = \frac{1}{x}$ is $u' = -\frac{1}{x^2}$ .
Apply Product Rule: The derivative of $v = e^{x}$ is $v' = e^{x}$ .
Substitute into Formula: Apply the product rule: $f'(x) = u'v + uv'$ .
Simplify Expression: Substitute $u'$ , $u$ , $v$ , and $v'$ into the formula: $f'(x) = (-\frac{1}{x^2})e^{x} + (\frac{1}{x})e^{x}$ .
Final Answer: Simplify the expression: $f'(x) = e^{x}(-\frac{1}{x^2} + \frac{1}{x})$ .
Final Answer: Simplify the expression: $f'(x) = e^{x}(-\frac{1}{x^2} + \frac{1}{x})$ .The final answer is $f'(x) = e^{x}(\frac{1}{x} - \frac{1}{x^2})$ .