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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 $\displaystyle f'(x)=\frac{a'}{a}$
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$\frac{1}{\frac{x\sqrt{x^2+3}}{\sqrt[3]{\left(x+9\right)^{2}}}}\frac{d}{dx}\left(\frac{x\sqrt{x^2+3}}{\sqrt[3]{\left(x+9\right)^{2}}}\right)$
Learn how to solve problems step by step online. Find the derivative using the product rule d/dx(ln((x(x^2+3)^1/2)/((x+9)^2/3))). 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{x\sqrt{x^2+3}}{\sqrt[3]{\left(x+9\right)^{2}}}} 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}. Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. Simplify \left(\sqrt[3]{\left(x+9\right)^{2}}\right)^2 using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals \frac{2}{3} and n equals 2.