Let y=sqrtxe^(x). (dy)/(dx)=

Identify Product Rule: To find the derivative of y with respect to x, we need to use the product rule because y is a product of two functions, sqrt(x) and e^(x).Define Functions: Let u = sqrt(x) and v = e^(x). Then, we need to find the derivatives of u and v with respect to x.Find Derivatives: The derivative of u with respect to x is (1/2)x^(-1/2) because the derivative of x^(1/2) is (1/2)x^(1/2 - 1).Apply Product Rule: The derivative of v with respect to x is e^(x) because the derivative of e^(x) with respect to x is e^(x).Substitute into Formula: Now, apply the product rule: (dy)/(dx) = u'(x)v(x) + u(x)v'(x).Simplify Expression: Substitute the derivatives and functions into the product rule: (dy)/(dx) = (1/2)x^(-1/2)e^(x) + sqrt(x)e^(x).Combine Terms: Simplify the expression: (dy)/(dx) = (1/2)x^(-1/2)e^(x) + x^(1/2)e^(x).Final Answer: Combine the terms: (dy)/(dx) = e^(x)((1/2)x^(-1/2) + x^(1/2)).The final answer is (dy)/(dx) = e^(x)((1/2)x^(-1/2) + x^(1/2)).

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Let $y=\sqrt{x} e^{x}$ . $\newline$ $\frac{d y}{d x}=$

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Algebra 2

Csc, sec, and cot of special angles

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Q. Let

y=\sqrt{x} e^{x}

\newline

\frac{d y}{d x}=

Identify Product Rule: To find the derivative of $y$ with respect to $x$ , we need to use the product rule because $y$ is a product of two functions, $\sqrt{x}$ and $e^{x}$ .
Define Functions: Let $u = \sqrt{x}$ and $v = e^{x}$ . Then, we need to find the derivatives of $u$ and $v$ with respect to $x$ .
Find Derivatives: The derivative of $u$ with respect to $x$ is $(1/2)x^{(-1/2)}$ because the derivative of $x^{(1/2)}$ is $(1/2)x^{(1/2 - 1)}$ .
Apply Product Rule: The derivative of $v$ with respect to $x$ is $e^{x}$ because the derivative of $e^{x}$ with respect to $x$ is $e^{x}$ .
Substitute into Formula: Now, apply the product rule: $(\frac{dy}{dx}) = u'(x)v(x) + u(x)v'(x)$ .
Simplify Expression: Substitute the derivatives and functions into the product rule: $(\frac{dy}{dx}) = (\frac{1}{2})x^{(-\frac{1}{2})}e^{x} + \sqrt{x}e^{x}$ .
Combine Terms: Simplify the expression: $(\frac{dy}{dx}) = (\frac{1}{2})x^{(-\frac{1}{2})}e^{(x)} + x^{(\frac{1}{2})}e^{(x)}.$
Final Answer: Combine the terms: $\frac{dy}{dx} = e^{x}\left(\frac{1}{2}x^{-\frac{1}{2}} + x^{\frac{1}{2}}\right)$ .
Final Answer: Combine the terms: $(dy)/(dx) = e^{x}((1/2)x^{(-1/2)} + x^{(1/2)})$ .The final answer is $(dy)/(dx) = e^{x}((1/2)x^{(-1/2)} + x^{(1/2)})$ .