Let y=sqrt(e^(x)). Find (dy)/(dx). Choose 1 answer: (A) (sqrt(e^(x)))/(2) (B) sqrt(e^(x)) (C) xsqrt(e^(x-1)) (D) (1)/(2sqrt(e^(x)))

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Let  y=sqrt(e^(x)). Find  (dy)/(dx). Choose 1 answer: (A)  (sqrt(e^(x)))/(2) (B)  sqrt(e^(x)) (C)  xsqrt(e^(x-1)) (D)  (1)/(2sqrt(e^(x)))

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Express Function in Terms of x: First, let's express the function y in terms of x using the square root and exponential notation. y = sqrt(e^x) This can also be written as: y = (e^x)^(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. (dy)/(dx) = d/dx[(e^x)^(1/2)]Calculate Derivative of Outer Function: The outer function is the square root function, which can be written as a power of 1/2. The derivative of a^b with respect to a is b*a^(b-1), so the derivative of (e^x)^(1/2) with respect to e^x is (1/2)*(e^x)^(-1/2). (dy)/(dx) = (1/2)*(e^x)^(-1/2) * 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. (dy)/(dx) = (1/2)*(e^x)^(-1/2) * e^xSimplify Expression: Now, we simplify the expression. Since (e^x)^(-1/2) is the same as 1/sqrt(e^x), we can rewrite the expression as: (dy)/(dx) = (1/2) * (1/sqrt(e^x)) * e^xFurther 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: (dy)/(dx) = (1/2) * (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: (dy)/(dx) = (1/2) * sqrt(e^x)Verify Correct Answer: Finally, we can see that the correct answer matches option (A): (dy)/(dx) = (sqrt(e^x))/(2)

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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Let y=ex y=\sqrt{e^{x}} y=ex​.\newlineFind dydx \frac{d y}{d x} dxdy​.\newlineChoose 111 answer:\newline(A) ex2 \frac{\sqrt{e^{x}}}{2} 2ex​​\newline(B) ex \sqrt{e^{x}} ex​\newline(C) xex−1 x \sqrt{e^{x-1}} xex−1​\newline(D) 12ex \frac{1}{2 \sqrt{e^{x}}} 2ex​1​

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