Let $y=\sqrt{e^{x}}$ . $\newline$ Find $\frac{d y}{d x}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{\sqrt{e^{x}}}{2}$ $\newline$ (B) $\sqrt{e^{x}}$ $\newline$ (C) $x \sqrt{e^{x-1}}$ $\newline$ (D) $\frac{1}{2 \sqrt{e^{x}}}$

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Algebra 2

Simplify the product of two radical expressions having same variable

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Q. Let

y=\sqrt{e^{x}}

\newline

Find

\frac{d y}{d x}

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

\sqrt{e^{x}}

\newline

(C)

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

\newline

(D)

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

Express Function in Terms of x: First, let's express the function $y$ in terms of $x$ using the square root and exponential notation. $\newline$ $y = \sqrt{e^x}$ $\newline$ This can also be written as: $\newline$ $y = (e^x)^{\frac{1}{2}}$
Find Derivative Using Chain Rule: Now, we need to find the derivative of $y$ with respect to $x$ . We will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. $\newline$ $\frac{dy}{dx} = \frac{d}{dx}\left[(e^x)^{\frac{1}{2}}\right]$
Calculate Derivative of Outer Function: The outer function is the square root function, which can be written as a power of $\frac{1}{2}$ . The derivative of $a^b$ with respect to $a$ is $b\cdot a^{(b-1)}$ , so the derivative of $(e^x)^{\frac{1}{2}}$ with respect to $e^x$ is $\frac{1}{2}\cdot(e^x)^{-\frac{1}{2}}$ . $\newline$ $\frac{dy}{dx} = \frac{1}{2}\cdot(e^x)^{-\frac{1}{2}} \cdot \frac{d}{dx}[e^x]$
Multiply Derivatives: The derivative of $e^x$ with respect to $x$ is simply $e^x$ . So we multiply the derivative of the outer function by the derivative of the inner function. $\newline$ $\frac{dy}{dx} = \frac{1}{2}\cdot(e^x)^{-\frac{1}{2}} \cdot e^x$
Simplify Expression: Now, we simplify the expression. Since $(e^x)^{-\frac{1}{2}}$ is the same as $\frac{1}{\sqrt{e^x}}$ , we can rewrite the expression as: $\newline$ $\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\sqrt{e^x}} \cdot e^x$
Further Simplification: We can simplify further by multiplying $e^x$ by the fraction. Since $e^x$ is the same as $\sqrt{e^x}$ squared, we get: $\newline$ $(dy)/(dx) = (1/2) \cdot (e^x / \sqrt{e^x})$
Divide and Simplify: Simplifying the expression by dividing $e^x$ by $\sqrt{e^x}$ gives us $\sqrt{e^x}$ , so we have: $\newline$ $\frac{dy}{dx} = \frac{1}{2} \cdot \sqrt{e^x}$
Verify Correct Answer: Finally, we can see that the correct answer matches option (A): $\frac{dy}{dx} = \frac{\sqrt{e^x}}{2}$