Find (d)/(dx)((2x)/(sqrt(x^(2)+3)))

Apply Quotient Rule: Use the quotient rule for differentiation, which is (v(u') - u(v')) / (v^2), where u = 2x and v = sqrt(x^2 + 3).Differentiate u: Differentiate u = 2x with respect to x to get u' = 2.Differentiate v: Differentiate v = sqrt(x^2 + 3) with respect to x. First, rewrite v as (x^2 + 3)^(1/2) to make it easier.Use Chain Rule: Now, use the chain rule to differentiate v. The derivative of the outer function with respect to the inner function (x^2 + 3) is (1/2)(x^2 + 3)^(-1/2). Then multiply by the derivative of the inner function, which is 2x. So, v' = (1/2)(x^2 + 3)^(-1/2) * 2x.Apply Quotient Rule: Now apply the quotient rule: (v(u') - u(v')) / (v^2). Plug in u, u', v, and v' to get [(sqrt(x^2 + 3))(2) - (2x)((1/2)(x^2 + 3)^(-1/2) * 2x)] / (sqrt(x^2 + 3))^2.Simplify Numerator: Simplify the numerator: [2sqrt(x^2 + 3) - (2x)(x/(sqrt(x^2 + 3)))].Simplify Denominator: Simplify the denominator: (sqrt(x^2 + 3))^2 is just x^2 + 3.Combine Terms: Now, combine the terms in the numerator: 2sqrt(x^2 + 3) - 2x^2/(sqrt(x^2 + 3)).Write Final Derivative: Write the final simplified derivative: [2sqrt(x^2 + 3) - 2x^2/(sqrt(x^2 + 3))] / (x^2 + 3).

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Find $\frac{d}{dx}\left(\frac{2x}{\sqrt{x^{2}+3}}\right)$

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Q. Find

\frac{d}{dx}\left(\frac{2x}{\sqrt{x^{2}+3}}\right)

Apply Quotient Rule: Use the quotient rule for differentiation, which is $(v(u') - u(v')) / (v^2)$ , where $u = 2x$ and $v = \sqrt{x^2 + 3}$ .
Differentiate $u$ : Differentiate $u = 2x$ with respect to $x$ to get $u' = 2$ .
Differentiate $v$ : Differentiate $v = \sqrt{x^2 + 3}$ with respect to $x$ . First, rewrite $v$ as $(x^2 + 3)^{\frac{1}{2}}$ to make it easier.
Use Chain Rule: Now, use the chain rule to differentiate $v$ . The derivative of the outer function with respect to the inner function $(x^2 + 3)$ is $(\frac{1}{2})(x^2 + 3)^{-\frac{1}{2}}$ . Then multiply by the derivative of the inner function, which is $2x$ . So, $v' = (\frac{1}{2})(x^2 + 3)^{-\frac{1}{2}} \cdot 2x$ .
Apply Quotient Rule: Now apply the quotient rule: $(v(u') - u(v')) / (v^2)$ . Plug in $u$ , $u'$ , $v$ , and $v'$ to get $\left[(\sqrt{x^2 + 3})(2) - (2x)\left(\frac{1}{2}(x^2 + 3)^{-\frac{1}{2}} \cdot 2x\right)\right] / (\sqrt{x^2 + 3})^2$ .
Simplify Numerator: Simplify the numerator: $[2\sqrt{x^2 + 3} - (2x)\left(\frac{x}{\sqrt{x^2 + 3}}\right)]$ .
Simplify Denominator: Simplify the denominator: $(\sqrt{x^2 + 3})^2$ is just $x^2 + 3$ .
Combine Terms: Now, combine the terms in the numerator: $2\sqrt{x^2 + 3} - \frac{2x^2}{\sqrt{x^2 + 3}}$ .
Write Final Derivative: Write the final simplified derivative: $\frac{2\sqrt{x^2 + 3} - \frac{2x^2}{\sqrt{x^2 + 3}}}{x^2 + 3}$ .