Find y^(') if y=ln ((sqrt(4+x^(2)))/(x)).

Identify function: Identify the function to differentiate. y = ln((sqrt(4+x^2))/x)Apply chain rule: Apply the chain rule for derivatives to the logarithmic function. Let u = (sqrt(4+x^2))/x y = ln(u) dy/dx = (1/u) * du/dxDifferentiate u: Differentiate u = (sqrt(4+x^2))/x using the quotient rule. u = (sqrt(4+x^2))/x du/dx = [(1/2)(4+x^2)^(-1/2) * 2x * x - (sqrt(4+x^2)) * 1] / x^2 = [(x/(sqrt(4+x^2))) - (sqrt(4+x^2))/x] / x

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Find $y'$ if $y=\ln \left(\frac{\sqrt{4+x^{2}}}{x}\right).$

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Calculus

Find derivatives of logarithmic functions

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Q. Find

y'

y=\ln \left(\frac{\sqrt{4+x^{2}}}{x}\right).

Identify function: Identify the function to differentiate. $\newline$ $y = \ln\left(\frac{\sqrt{4+x^2}}{x}\right)$
Apply chain rule: Apply the chain rule for derivatives to the logarithmic function. $\newline$ Let $u = \frac{\sqrt{4+x^2}}{x}$ $\newline$ $y = \ln(u)$ $\newline$ $\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}$
Differentiate $u$ : Differentiate $u = \frac{\sqrt{4+x^2}}{x}$ using the quotient rule. $\newline$ $u = \frac{\sqrt{4+x^2}}{x}$ $\newline$ $\frac{du}{dx} = \frac{\left(\frac{1}{2}(4+x^2)^{-\frac{1}{2}} \cdot 2x \cdot x - \sqrt{4+x^2} \cdot 1\right)}{x^2}$ $\newline$ $= \frac{\left(\frac{x}{\sqrt{4+x^2}} - \frac{\sqrt{4+x^2}}{x}\right)}{x}$