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

Apply Quotient Rule: Apply the quotient rule for differentiation, which states that the derivative of a function f(x)/g(x) is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2. Let f(x) = e^x and g(x) = sqrt(x) = x^(1/2). We need to find f'(x) and g'(x).Find f'(x): Find the derivative of f(x) = e^x with respect to x. The derivative of e^x with respect to x is e^x. So, f'(x) = e^x.Find g'(x): Find the derivative of g(x) = x^(1/2) with respect to x. Using the power rule, (d/dx)(x^n) = nx^(n-1), we get: g'(x) = (1/2)x^(1/2 - 1) = (1/2)x^(-1/2) = (1/2)(1/sqrt(x)).Apply Quotient Rule: Apply the quotient rule using the derivatives from steps 2 and 3. (d/dx)((e^x)/(sqrt(x))) = (e^x * (1/2)(1/sqrt(x)) - e^x * (1/sqrt(x)))/(x^(1/2))^2.Simplify Expression: Simplify the expression. (d/dx)((e^x)/(sqrt(x))) = (e^x/(2sqrt(x)) - e^x/sqrt(x))/(x).Combine Terms: Combine the terms in the numerator and simplify the denominator. (d/dx)((e^x)/(sqrt(x))) = (e^x/(2sqrt(x)) - 2e^x/(2sqrt(x)))/(x^(1/2))^2. (d/dx)((e^x)/(sqrt(x))) = (-e^x/(2sqrt(x)))/(x).Simplify Denominator: Simplify the complex fraction. (d/dx)((e^x)/(sqrt(x))) = -e^x/(2x^(3/2)).Simplify Complex Fraction: Notice that none of the answer choices match the simplified derivative we found. There must be a mistake in the previous steps. Let's go back and check our calculations.

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Find

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

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Choose

1

answer:

\newline

(A)

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

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(B)

2 \sqrt{x} e^{x}

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(C)

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

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(D)

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

Apply Quotient Rule: Apply the quotient rule for differentiation, which states that the derivative of a function $f(x)/g(x)$ is given by $(f'(x)g(x) - f(x)g'(x))/(g(x))^2$ . Let $f(x) = e^x$ and $g(x) = \sqrt{x} = x^{(1/2)}$ . We need to find $f'(x)$ and $g'(x)$ .
Find $f'(x)$ : Find the derivative of $f(x) = e^x$ with respect to $x$ . The derivative of $e^x$ with respect to $x$ is $e^x$ . So, $f'(x) = e^x$ .
Find $g'(x)$ : Find the derivative of $g(x) = x^{\frac{1}{2}}$ with respect to $x$ . Using the power rule, $\frac{d}{dx}(x^n) = nx^{n-1}$ , we get: $g'(x) = \frac{1}{2}x^{\frac{1}{2} - 1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2}(\frac{1}{\sqrt{x}})$ .
Apply Quotient Rule: Apply the quotient rule using the derivatives from steps $2$ and $3$ . $\newline$ $(\frac{d}{dx})\left(\frac{e^x}{\sqrt{x}}\right) = \frac{e^x \cdot (\frac{1}{2})(\frac{1}{\sqrt{x}}) - e^x \cdot (\frac{1}{\sqrt{x}})}{(x^{\frac{1}{2}})^2}$ .
Simplify Expression: Simplify the expression. $(\frac{d}{dx})(\frac{e^x}{\sqrt{x}}) = \frac{e^x}{2\sqrt{x}} - \frac{e^x}{\sqrt{x}})\over{x}$ .
Combine Terms: Combine the terms in the numerator and simplify the denominator. $\newline$ (\frac{d}{dx})\left(\frac{e^x}{\sqrt{x}}\right) = \frac{\frac{e^x}{$2$\sqrt{x}} - \frac{$2$e^x}{$2$\sqrt{x}}}{x^{\frac{$1$}{$2$}}^$2$}.$\newline(\frac{d}{dx})\left(\frac{e^x}{\sqrt{x}}\right) = \frac{-\frac{e^x}{\(2$\sqrt{x}}}{x}.
Simplify Denominator: Simplify the complex fraction. \((\frac{d}{dx})\left(\frac{e^x}{\sqrt{x}}\right) = -\frac{e^x}{2x^{\frac{3}{2}}}.
Simplify Complex Fraction: Notice that none of the answer choices match the simplified derivative we found. There must be a mistake in the previous steps. Let's go back and check our calculations.