Find the derivative using the quotient rule $\frac{d}{dx}\left(\ln\left(\frac{x^3\sqrt{3e^x\cdot x^2}}{\left(x+1\right)^2\left(x-1\right)^3}\right)\right)$

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

Learn how to solve differential equations problems step by step online. Find the derivative using the quotient rule d/dx(ln((x^3(3e^xx^2)^1/2)/((x+1)^2(x-1)^3))). The power of a product is equal to the product of it's factors raised to the same power. When multiplying exponents with same base you can add the exponents: \sqrt{3}x^3e^{\frac{1}{2}x}x. 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}. Divide fractions \frac{1}{\frac{\sqrt{3}x^{4}e^{\frac{1}{2}x}}{\left(x+1\right)^2\left(x-1\right)^3}} with Keep, Change, Flip: a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}.