Find (d)/(dx)(ln(x)e^(x)). Choose 1 answer: (A) (e^(x))/(x) (B) (e^(x))/(x)+ln(x) (C) (1)/(x)+e^(x) (D) e^(x)((1)/(x)+ln(x))

Apply product rule: Use the product rule for derivatives: (d/dx)(u*v) = u'v + uv'.Identify u and v: Let u = ln(x) and v = e^(x). Now find u' and v'.Find derivatives: The derivative of ln(x) is 1/x, so u' = 1/x.Use product rule: The derivative of e^(x) is e^(x), so v' = e^(x).Simplify expression: Now apply the product rule: (d/dx)(ln(x)e^(x)) = (1/x)e^(x) + ln(x)e^(x).Simplify the expression: (d/dx)(ln(x)e^(x)) = e^(x)/x + ln(x)e^(x).

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Find
(d)/(dx)(ln(x)e^(x)).
Choose 1 answer:
(A)
(e^(x))/(x)
(B)
(e^(x))/(x)+ln(x)
(C)
(1)/(x)+e^(x)
(D)
e^(x)((1)/(x)+ln(x))

Find $\frac{d}{d x}\left(\ln (x) e^{x}\right)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{e^{x}}{x}$ $\newline$ (B) $\frac{e^{x}}{x}+\ln (x)$ $\newline$ (C) $\frac{1}{x}+e^{x}$ $\newline$ (D) $e^{x}\left(\frac{1}{x}+\ln (x)\right)$

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Calculus

Find derivatives of using multiple formulae

Full solution

Q. Find

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

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

\frac{e^{x}}{x}+\ln (x)

\newline

(C)

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

\newline

(D)

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

Apply product rule: Use the product rule for derivatives: $(\frac{d}{dx})(u\cdot v) = u'v + uv'$ .
Identify $u$ and $v$ : Let $u = \ln(x)$ and $v = e^{x}$ . Now find $u'$ and $v'$ .
Find derivatives: The derivative of $\ln(x)$ is $\frac{1}{x}$ , so $u' = \frac{1}{x}$ .
Use product rule: The derivative of $e^{x}$ is $e^{x}$ , so $v' = e^{x}$ .
Simplify expression: Now apply the product rule: $(\frac{d}{dx})(\ln(x)e^{(x)}) = (\frac{1}{x})e^{(x)} + \ln(x)e^{(x)}.$
Simplify expression: Now apply the product rule: $(\frac{d}{dx})(\ln(x)e^{(x)}) = (\frac{1}{x})e^{(x)} + \ln(x)e^{(x)}.$ Simplify the expression: $(\frac{d}{dx})(\ln(x)e^{(x)}) = \frac{e^{(x)}}{x} + \ln(x)e^{(x)}.$