Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(\ln\left(\frac{x\sqrt{x^2+1}}{\sqrt[3]{\left(x+9\right)^{2}}}\right)\right)$

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$\frac{d}{dx}\left(\ln\left(x\right)+\frac{1}{2}\ln\left(x^2+1\right)-\frac{2}{3}\ln\left(x+9\right)\right)$

Learn how to solve logarithmic differentiation problems step by step online. Find the derivative using logarithmic differentiation method d/dx(ln((x(x^2+1)^(1/2))/((x+9)^(2/3)))). Simplify the derivative by applying the properties of logarithms. The derivative of a sum of two or more functions is the sum of the derivatives of each function. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}.